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DYNAMIC ANALYSIS OF FOUR STORY BUILDING By Kalpesh Parikh Pursuing Master of Science, Syracuse University

Term Report Submitted in partial fulfillment of the requirements for the course requirement of Master of Science in Civil Engineering in the Graduate School of Syracuse University 10th May 2010

Approved ______________________________ Professor Eric M. Lui Grade___________________________________

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Acknowledgment

My deepest gratitude goes to Dr. Eric M. Lui, Assistant professor, for his continuous and constructive

advice and follow-up. His successive advisories and comments were the pillars in my every step during

the analysis process of the project. I am thankful to him for the fact that he has inspired and helped me to

know about the Dynamic & Earthquake Engineering.

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Table of Content S.No.

Title

Page No

Acknowledgement Table of contents List of Figures List of Tables 1

Introduction

6

2

Loads and Functions

17

3

Modeling and Analysis Description

19

4

Load Models

23

5

Analysis

25

6

Conclusions

46

7

References

47

4

LIST OF FIGURES Figure 1 Original Plan View of the Four Story Building Figure 2 Elevation View of the Four Story Building (Y-Z axis) Figure 3 Seattle Spectra (Response Spectrum Curve) Figure 4 Comparison of Stiffness ratio for Model 2 and Model 3 Figure 5 Time History Spectra- LACCO NOR earthquake record (obtained using SAP 2000) Figure 6 Showing Beam & Exterior Column Arrangement Figure 7 Showing Beam & Interior Column Arrangement Figure 8 Model with Dead load Figure 9 Model with Live Load Figure 10 Showing displacement under Seattle Spectra-Model 1 Figure 11 Showing shear force under Seattle Spectra- Model 1 Figure 12 Showing bending moment (at grid line 2) under Seattle Spectra-Model1 Figure 13 Showing Displacements under Seattle Spectra-Model 2 Figure 14 Showing shear force under Seattle Spectra- Model 2 Figure 15 Showing bending moment (at grid line 1) under Seattle Spectra-Model2 Figure 16 Showing displacement under Seattle Spectra-Model 3 Figure 17 Showing shear force under Seattle Spectra- Model 3 Figure 18 Showing bending moment (at grid line 1) under Seattle Spectra-Model3 Figure 19 Showing bending moment (at grid line 2) under Seattle Spectra-Model3 Figure 20 Comparison of Displacement for 3 different models Figure 21 Showing displacement under LACCO NOR earthquake record-Model 4 Figure 22 Comparison of Joint displacement under LACCO NOR earthquake record-Model 4 Figure 23 Comparison of Joint displacement under LACCO NOR earthquake record-Model 4 Figure 24 Model 5 Showing Rubber Isolator. Figure 25 Comparison of Joint displacement under LACCO NOR earthquake record-Model 5 Figure 26 Comparison of Joint displacement under LACCO NOR earthquake record-Model 5 Figure 27 Comparison of Joint Vs Base Shear under LACCO NOR earthquake record-Model 5 Figure 28 Layout of Link Element Figure 29 Isolator Deformations –Model 5-Link Set 1 Figure 30 Isolator Deformations –Model 5-Link Set 2

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LIST OF TABLES Table 1 : Function of Response Spectrum Function-IBC 2006 Table 2 : Calculation of Seismic Lateral Force Table 3: Floor height description for model 2 and model 3 Table 4: Shear wall dimensioning Table 5: Summary of Stiffness for Beams and Columns for Model 2 and Model 3(Soft Story): Table 6: Effect of Stiffness due to soft story model (ht variation) Table 6a: Summary of Dead Load (IBC, minimum design dead load (Table C3-1)) Table 7: Summary of Live Load (Obtained from the IBC minimum uniformly distributed live load (Table 4-1) and shown below) Table 8 : Comparison Tables and Result Obtained for 3 models Table 9: Comparison Calculation for finding % reduction of displacement due to soft story Table 10: Comparison of base reaction due to all 3 model & % reduction of base reaction due to soft story Table 11: Response Spectrum Analysis Model 1 Table 12: Response Spectrum Analysis Model 2 Table 13: Response Spectrum Analysis Model 3 Table 14: Comparison of effect of soft story in RSA Table 15: Modal Periods and Frequencies for LACCO NOR earthquake-Model 4 Table 16: Base Reaction for LACCO Spectra-Model 5 Table 17: Modal period and frequencies-Model 5 Table 18: Comparison of Period of Model 4 & Model 5 Table 19: Comparison of Base Reaction of Model 4 & Model 5

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1 Introduction All real physical structures behave dynamically when subjected to loads or displacements. The additional inertia forces, from Newton’s second law, are equal to the mass times the acceleration. If the loads or displacements are applied very slowly, the inertia forces can be neglected and a static load analysis can be justified. Hence, dynamic analysis is a simple extension of static analysis. In addition, all real structures potentially have an infinite number of displacements. Therefore, the most critical phase of a structural analysis is to create a computer model with a finite number of massless members and a finite number of node (joint) displacements that will simulate the behavior of the real structure. Therefore based on the complexity involved in the hand calculation an computer model is made using SAP 2000 based on the model, simulate the behavior of the real structure under a dynamic loading .To accomplish the good understanding of dynamic behavior I selected a four story concrete building, located in Seattle, Washington (seismic zone 3) below are the plan showing how the floor plan looks like for Stories 1 to 4.

Figure 1 Original Plan View of the Four Story Building

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Figure 2 Elevation View of the Four Story Building (Y-Z axis)

Seismic weight at various floors: For a Warehouse, the design load should include a minimum of 25% of the live load. No live load is to be considered for roof. Hence, the effective weight at all floors, except at the roof will be 140  0.25125   171.25 Psf, and the effective weight for roof will be 140 psf. The Plan area is 48 ft x 96ft = 4608 ft2. Hence Seismic weights of various levels are: W1 = W2 = W3 = 1 st, 2nd & 3rd Story weight, W1 = W2 = W3 = 4608 x 0.17125 = 789.1 Kips & W4 = 4608 x 0.140 = 645.1 Kip The total Seismic weight of the building is then W = 789.1 x 3 + 645.1 = 3012.4 Kip

Fundamental Period of Building: T = Ct * hn3/4 Where: Ct = 0.030 (for reinforcing concrete moment-resisting frame) hn = 48 ft (total height of the building) T = 0.030* 483/4 = 0.55 sec Occupancy Importance Factor: Warehouse (SUG) = I = 1 and Occupancy importance factor, IE = 1

8 TABLE 1: Function - Response Spectrum -IBC2006

Seattle Spectra for Zip Code 94704 70

Spectral acceleration Sa (inch/sec^2)

60

Seattle Spectra

50 40 30 20

Period( sec)

Accel (in/sec^2)

0

26.1607612

0.076841

65.401903

0.384205

65.401903

0.6

41.8796098

0.8

31.4097154

1

25.127753

1.2

20.939821

1.4

17.9484088

1.6

15.7048416

1.8

13.9598592

2

12.5638926

2.5

10.0511012

3

8.3759284

3.5

7.1793764

4

6.2819302

4.5

5.583963

5

5.0255506

5.5

4.568697

6

4.1879642

6.5

3.8658032

7

3.5896882

7.5

3.3503778

8

3.1409812

8.5

2.7823054

10 0 0

2

4

6 Period (sec)

8

Figure 3 Seattle Spectra (Response Spectrum Curve)

10

12

9

Mapped Response Spectral Acceleration: (Use of SAP 2000) as shown above spectra: Computer I/P: Code Selection: IBC 2006 (IBC 2010 not available) Soil Class: B for rock Damping: 0.05 Zip Code: 94704 Results: Short Period (T = 0.2 sec)

Ss = 3.046673g

Long Period (T = 1 sec)

S1= 1.170548 g

Site Class = B for rock Site coefficient = Fa = 1

SDS = 2.031115

Site coefficient = Fv = 1

SD1 = 0.780365

Soil Modified Response Spectral Acceleration: SMS = Fa Ss = 3.046673 SM1 = Fv S1 = 1.170548 Design Response Spectral Acceleration: SDS = 2* 3.046673 / 3

SDS = 2.031115 (Same as obtain from SAP 2000)

SD1 = 2* 1.170548 / 3

SD1 = 0.780365 (Same as obtain from SAP 2000)

Response Modification Factor: R= 8 for Special Reinforced Concrete Moment Frame (obtained using table 12.2-1 Design coefficient and factors for seismic force resisting system ASCE 7-05) Seismic Design Category = D Seismic Coefficient: Cs = SDS*IE/R = 0.253889 Check minimum value for Cs : Cs

0.044 * SD1 * IE = 0.03433606……………… Good!

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S1*0.5*IE/R = 0.073159……………………… Good!

Cs Then

Cs = 0.253889

Base Shear Force: V = Cs * W V= 0.253889 * 3012.4 V = 764.815 Kip Where: V = Seismic Base Shear. Cs = Seismic Response Coefficient. W = Seismic weight of the structure that includes the dead weight and any permanent loading in this case it also includes 25% of live load as per IBC code provision Vertical Force Distribution: F= For, T = 0.55 sec > 0.5 sec K = 1.025 (by Interpolation) Table2 : Calculation of Seismic Lateral Force Level 4 3 2 1

hx (ft) 48 36 24 12

Wx (Kip) 645.1 781.1 781.1 781.1

hxk (ft) 52.88 39.37 25.98 12.77

Wx h x k (Kip-ft) 34113 30752 20293 9974 95132

Fx (Kip) 274.25 247.23 163.145 80.186

Vx (Kip) 274.25 521.48 684.625 764.811

Mx (Kip-ft) 3291 9548.76 17764.26 30604.02

Overturning Moment: Mx = = 1 (for top 10 story)

(as calculated above in table)

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Story Drift and Lateral Displacement: Both strength and stiffness need to be considered in the design of special moment frames. According to ASCE 7, special moment frames are allowed to be designed for a force reduction factor of R = 8. That is, they are allowed to be designed for a base shear equal to one-eighth of the value obtained from an elastic response analysis. Moment frames are generally flexible lateral systems; therefore, strength requirements may be controlled by the minimum base shear equations of the code. Base shear calculations for long-period structures, has been checked and may govern the strength requirements of special moment frames. The allowable story drift, ∆a = 0.025 hx = 3.6 inch (where hx is the story height) Stiffness Computation: Kcol = 12*E*I / L3 Econc = 3600 Ksi = modulus of elasticity of concrete LCol = 12’-0” 1st Story and 2nd Story Stiffness Computation: a) Exterior Column : 12” x 20” b) Interior Column : 12” x 24” a) Exterior Column for First Story Kextcol.1= 115.47 Kip/inch b) Interior Column for First Story Kintcol.1= 200 Kip/inch

Iextcol = 8000 in4 Iintcol = 13824 in4 a) Exterior Column for Second Story Kextcol.2= 115.47 Kip/inch b) Interior Column for Second Story Kintcol.2= 200 Kip/inch

Total Stiffness: KTotal Col 1= 18*115.47 + 9* 200 = 3878.46 Kip/inch KTotal Col 2= 18*115.47 + 9* 200 = 3878.46 Kip/inch 3rd Story and 4th Story Stiffness Computation: a) Exterior Column : 12” x 16” b) Interior Column : 12” x 20”

Iextcol = 4096 in4 Iintcol = 8000 in4

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a) Exterior Column for Third Story

a) Exterior Column for Forth Story

Kextcol.3= 59.25 Kip/inch b) Interior Column for Third Story

Kextcol.4= 59.25 Kip/inch b) Interior Column for Forth Story

Kintcol.3= 115.47 Kip/inch

Kintcol.4= 115.47 Kip/inch

Total Stiffness: KTotal Col 3= 18*59.25 + 9* 115.47 = 2105.73 Kip/inch KTotal Col 4= 18*59.25 + 9* 115.47 = 2105.73 Kip/inch Beams Stiffness Kbeam= 3*E*I / L3 Econc = 3600 Ksi = modulus of elasticity of concrete Lbeam = 24’-0” 1st Story to 4th Story Stiffness Computation: Beam Size: 20” x 20”

Ibeam = 13333.33 in4

Kbeam= 6.028 Kip/inch Total Stiffness: KTotal Beam 1= 42*6.028 = 253.176 Kip/inch KTotal Beam 1= KTotal Beam 2= KTotal Beam 3= KTotal Beam 4= 253.176 Kip/inch Material Properties Rebar: Reinforcement for Beams and Columns Type: A615Gr60 Weight per unit volume = 0.49 Kip/ft3

Fy = 60 Ksi Fu = 90 Ksi

Modulus of Elasticity (E) = 29000 Ksi Concrete: Use for Beams, Columns, Floors and Wall Concrete compressive Strength Fc’ = 4000 Psi LWC Shear Reduction Factor = 0.8

Modulus of Elasticity (E) = 3600 Ksi Weight per unit volume = 0.15 Kip/ft3

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***Use of light weight concete(LWC) is made for columns, beams & floors & Concrete use for the Shear Walls use of Normal Weight concrete is made** Rubber Isolator: Isolated Pad for Supports Weight of each isolator pad = 32.2 lb (too small but mass of base slab is provided above it) Vertical Axial Stiffness = 10000 k/in

Initial Shear Stiffness in each direction = 10 k/in

Shear Yield Force in each direction = 5 kips Ratio of Post Yield Shear Stiffness to Initial shear stiffness =0.2 Soil Type : Site is located in the Seattle, Washington as per IBC site class definition Site Class : B Soil Profile Name: Rock Seismic Zone Factor : The seismic zone factor z is computed by referring a Seismic zone map where seattle region falls under Zone 3 , Z=0.3 Description about dimensioning Floors: Floor Dimension: Rectangular plan 48’-0” x 96’-0” (same for each story 1 to 4). Please coordinate with Plan Drawing. Floor slab used for the building is shell plate thin element of thickness 10 inch both in membrane and bending. Floor to Floor height: Table 3: Floor height description for model 2 and model 3 Floor Ground-1st Floor 1st to 2nd Floor 2nd to 3rd Floor 3rd to 4th Floor

Model 2 (Story) ht in ft 12’-0” 12’-0” 12’-0” 12’-0”

Model 3 (Soft Story) ht in ft 11’-0” 11’-0” 11’-0” 15’-0”

Shear Wall: Shear wall is being considered in Model 3. For that the material properties is being changed from LWC to NWC. Thickness of the wall considered 12” thick. Its placement in oriented by following 3-dimensional co-ordinate .To give revelation can be co-ordinate with model and plan.

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Table 4: Shear wall dimensioning Name Wall Panel A-B Wall Panel 1-2 Wall Panel H-J Wall Panel 2-3

Size (ft) 12’-0” x 1’-0” x 48’-0” 12’-0” x 1’-0” x 48’-0” 12’-0” x 1’-0” x 48’-0” 12’-0” x 1’-0” x 48’-0”

Start Co-ordinate(ft) -48’-0”, 24’-0”, 48’-0” -48’-0”, -24’-0”, 48’-0” 48’-0”, - 24’-0”, 48’-0” 48’-0”, 0’-0”, 48’-0”

End Co-ordinate(ft) -36’-0”, 24’-0”, 48’-0” -48’-0”, 0’-0”, 48’-0” -36’-0”, -24’-0”, 48’-0” 48’-0”, 24’-0”, 48’-0”

Stiffness Computation For Soft Story: Kcol = 12*E*I / L3 Econc = 3600 Ksi = modulus of elasticity of concrete LCol = 11’-0” ( for 1 to 3rd Story)

LCol = 12’-0” ( for 4th Story)

1st Story and 2nd Story Stiffness Computation: c) Exterior Column : 12” x 20” d) Interior Column : 12” x 24” c) Exterior Column for First Story Kextcol.1= 150.26 Kip/inch d) Interior Column for First Story Kintcol.1= 259.65 Kip/inch

Iextcol = 8000 in4 Iintcol = 13824 in4 c) Exterior Column for Second Story Kextcol.2= 150.26 Kip/inch d) Interior Column for Second Story Kintcol.2= 259.65

Kip/inch

Total Stiffness: KTotal Col 1= 18*150.26+ 9* 259.65 = 5041.53 Kip/inch KTotal Col 2= 18*150.26 + 9* 259.65 = 5041.53 Kip/inch 3rd Story and 4th Story Stiffness Computation: c) Exterior Column : 12” x 16” d) Interior Column : 12” x 20” c) Exterior Column for Third Story Kextcol.3= 76.93 Kip/inch d) Interior Column for Third Story Kintcol.3= 150.26. Kip/inch

Iextcol = 4096 in4 Iintcol = 8000 in4 c) Exterior Column for Forth Story Kextcol.4= 30.34 Kip/inch d) Interior Column for Forth Story Kintcol.4= 59.26 Kip/inch

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Total Stiffness: KTotal Col 3= 18*76.93+ 9* 150.26 = 2737.08 Kip/inch KTotal Col 4= 18*30.34 + 9* 59.26 = 1079.46 Kip/inch Beams Stiffness Kbeam= 3*E*I / L3 Econc = 3600 Ksi = modulus of elasticity of concrete Lbeam = 24’-0” 1st Story to 4th Story Stiffness Computation: Beam Size: 20” x 20”

Ibeam = 13333.33 in4

Kbeam= 6.028 Kip/inch Total Stiffness: KTotal Beam 1= 42*6.028 = 253.176 Kip/inch KTotal Beam 1= KTotal Beam 2= KTotal Beam 3= KTotal Beam 4= 253.176 Kip/inch

Table 5: Summary of Stiffness for Beams and Columns for Model 2 and Model 3(Soft Story): Floor No. 1 2 3 4

Model 2 (Uniform ht Story) KTotal Beam KTotal Col Λ 253.176 3878.46 0.0653 253.176 3878.46 0.0653 253.176 2105.73 0.120 253.176 2105.73 0.120

Model 3 (Soft Story) KTotal Beam KTotal Col λ 253.176 5041.53 0.0502 253.176 5041.53 0.0502 253.176 2737.08 0.0924 253.176 1079.46 0.2345

Where, λ = KTotal Beam / KTotal Col Remark: We can see because increase in ht at the 4th level the columns stiffness for each Floor rearrange as shown below

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Table 6: Effect of Stiffness due to soft story model (ht variation) Floor No.

Model 3 (Soft Story) compare with Model2 Comparison of stiffness (Model 3 compared to Model 2)

1 2 3 4

29.9879 % increase 29.9879 % increase 29.9879 % increase 48.73702% decrease

Figure 4: Comparison of Stiffness ratio for Model 2 and Model 3

Comparasion of Stiffness ratio for 2 Models

Uniform Ht. Story Soft Story

Floor

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0

0.05

0.1

0.15

0.2

Stiffness of Beam/Stiffness of Column

0.25

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2 Loads and Functions This building is to be analyzed for dead, live, and earthquake functional load. Dead load: Dead Loads are the weights of materials, equipments or components that remains constant throughout the structure's life. In the project it includes weight of the materials and components which are used for floor, ceiling, partitioning and roof.. Table 6a: Summary of Dead Load (IBC, minimum design dead load (Table C3-1)) Type Total load on each floor

Description Loads from IBC (psf) Dead load estimated due to 140 (floor slab, beam, half wt. of the column above and below the floor partion wall)

Live Load : Which is weight which is superimposed on, or temporarily attached to, a structure (people, machinery and equipment, furniture, appliances, etc.). Table 7: Summary of Live Load (Obtained from the IBC minimum uniformly distributed live load (Table 4-1) and shown below) Floors Description Uniform (psf) st nd rd 1 , 2 & 3 Floor Warehouse 125 psf Roof Warehouse 50 psf

Response-Spectrum Functions: Design Spectra are not uneven curves; the spectra are intended to be the average of many earthquakes. This approach allow us obtain an maximum value of Umax. For that reason to obtain conservative study about earthquake analysis I selected IBC 2006 building code for design spectra to obtain. Where we can define, a response spectrum function is a series of digitized pairs of structural period and corresponding pseudo-spectral acceleration values. Based on the function Response Spectrum Curve is generated with respect of I/P data assigned to computer and we obtain an o/p of digitized points of pseudo-acceleration response versus period of structure. As explained above a I/P data was assigned to SAP 2000 software and we obtain o/p as shown in figure 3. Time-History Functions: The response history analysis is presented for an arbitrary structural configuration and very handful for multi story building with a unsymmetrical plan. It is mainly devoted to a single

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component of ground motion, typically one of the horizontal components. Combining the structural response determined from such independent analysis for each excitation components gives the response of linear system to multi-component excitation. Based on that I picked up LACCO NOR File from SAP 2000 this is what we get as an I/P. Figure 5: Time History Spectra- LACCO NOR earthquake record (obtained using SAP 2000)

Time History Spectra- For LACCO 10 Time History Spectra- For LACCO

8 6 4 Psuedo acceleration 2 in/sec2 0 0

10

20

30

40

-2 -4 -6 -8 Time (s)

50

60

70

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3 Modeling and Analysis Description Preparation of 5 models is performed and they are discussed as below: 1. Model 1 3-d four Story building without shear wall. And performed Response Spectrum Analysis for the model. 2. Model 2 3-d four Story building with shear wall. And performed Response Spectrum Analysis for the model. 3. Model 3 3-d four Story (soft story, 1.e. floor ht. variation was performed) building with shear wall. And performed Response Spectrum Analysis for the model. 4. Model 4 3-d four Story building with shear wall. And performed Time History Analysis for the model. 5. Model 5 3-d four Story building (here rubber isolator and mass slab is provided) with shear wall. And performed Time History Analysis for the model. Discussion about Modeling and Analysis I/p: 1. Rectangular 3-d frame of 96’-0” x 48’-0” x 48’-0” was generated. 2. Material: Concrete was defined for the building except shear wall material of concrete used is NWC and for shear wall 3. Frame Properties: Beams and columns were grouped into i. Beam ii. External Column 1st level & 2nd level iii. External Column 3rd level & 4th level iv. Internal Column 1st level & 2nd level v. Internal Column 3rd level & 4th level

Figure 6 Showing Beam & Exterior Column Arrangement

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Figure7 Showing Beam & Interior Column Arrangement 4. Frame Meshing was at joints and at intersection with frame 5. Area Section was defined Advantage: Shell element has it own local co-ordinate system. The shell element always activates all 6 Degree of freedom at each connected joints. Results for internal forces and moments are good. i. Floor: Plate thin shell element was defined reason the plate bending behavior includes two–way, out-of plane. Plate rotational stiffness components and a translation stiffness component in the direction normal to the plane of the element. By default it neglects shearing deformation and it is recommended to use plate structure for floor slab. ii. Shear wall: Use of Shell thin element. reason why we use this because when we compute an analysis to RSA if we provide thin panel element then the peak value of the shear stress will be good estimation of the damage index (For story drift calculation) 6. Assigned Joints Constraints: Assigning of diaphragm constraint causes all of its constraint joint to move together as a planar diaphragm which is rigid against membrane deformation. Concrete floors which has very high in- plane stiffness. Hence diaphragm reduces error in plane stiffness in floor. 7. Assign joint restraints at base level Z = 0 for all model fixed support except for Model 5link/support properties Isolator are provided. 8. Assigning area loads uniform shell, defining loads as shown in table mentioning dead load and live load.

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9. Define Functions: For Model 1 to 3 we analyzed using Response Spectrum IBC 2006 and for Model 4 & 5 we analyzed using Lacco Time History Function. As we have discuss how we obtain spectra using SAP 2000. These loads are used for applying ground accelerations in response-spectrum analyses and are used as starting load vectors for Ritzvector analysis. Here the acceleration load is computed for each joint and element and summed over the whole structure. Acceleration load for the analysis are transformed from global co-ordinate system to local co-ordinate system. 10. Addition of Load Cases: Spectra generated from function will now be added to the load cases Model 1 to 3 - Response Spectrum IBC 2006 - Spectra generated “Seattle Spectra” Now we select CQC method of modal combination because it is the most conservative method that is used to estimate a peak value displacement or force within a structure This approach assumes that the maximum modal values for all modes occurs at the same point in time.CQC method takes into account the statistical coupling between closely Space mode caused by modal damping. Key thing is if damping is zero it degenerates to SRSS method. 11. For Directional combination SRSS method is better because for each displacement force or stress quantity in the structure, modal combination produces single positive results for each direction of acceleration the value for a given response combine to produce single positive results. SRSS methods combine the response for different direction of loading. 12. Now assigning the Seattle spectra in X (U1) and Y (U2) direction here lot of study has been conducted about assigning the earthquake motion from all possible direction. Orthogonal effects in spectral analysis: The member in the structure should be designed for 100% of prescribed seismic forces in one direction plus 30% of prescribed forces in perpendicular direction. Here it can be reasonable to assume that motion that takes place during an earthquake has one principal direction or during a finite period of time when maximum when maximum ground acceleration occurs, a principal direction exists. But exact nature of 3dimensional wave propagation is not known. Based on the assumption, we can conclude that a structure must resist a major earthquake motion of magnitude of “ X” for all possible angles “Ѳ” and at the same point in time resist earthquake motion at 90 degree to the angle “Ѳ”. For the Model with RSA I have tested with 100% of IBC 2006 called “Seattle Spectra” in Y-(U2) direction and 30% of IBC 2006 called “Seattle Spectra” in X-(U1) direction. The Model is also tested vice versa and notice the difference in displacement. For a structure of importance and estimate over conservative analysis we can multiply by the factor safety to the spectra so that it reads out analysis for higher values and give more conservative results then needed. 13. Modal Load Case Modification here we have to decide what modes we have to put for the analysis no. of modes are not arbitrary it depends on D.O.F but we for this building we have many D.O.F we don’t want to put the many no’s of D.O.F it is trail to try with 20 and 30 and see the Modal participating mass ratios if it reaches to 95% then it will be

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reasonable analysis to accept with it. Even the importance of mentioning “Types of mode area” there are 2 modes of area. Eigenvector Analysis and Ritz vector Analysis it important to know which gives better results. Eigen vector analysis determine the undamped free vibration mode shapes and frequencies of the structure, but lot of research have been conducted the natural free vibration mode shapes are not the best basis for a mode superposition analysis of structures subjected to dynamic loads. Ritz vectors yield more accurate results than eigenvector. Because ritz vectors gives better results because taking into account the spatial distribution of dynamic loading. Knowing this we can proceed with applying accelerated load in global co-ordinate system in X-direction and Y-direction. 14. Model 4 &5 analysis I/p explanation the Lacco Time History data obtained from SAP 2000 file it is just a record of single earthquake the data obtained it is applied to the structure using local co-ordinate, here the orthogonality will not come in role, the importance of time history analysis which super cedes the RSA the input of Lacco Time History data assigned , for SAP 2000 it is possible to perform a large amount of dynamic analyses at various angles of input where we can check all points for critical earthquake direction. Here In Model 5 in co-operated the non linear analysis, because the advantage compare to RSA we have that we can perform non linear analysis in THA. RSA has limitation in nonlinear analysis 15. Model 5, to perform non linear analysis Here new load case is defined in the name of Grav this is restricted to the dead load only the manner in which applied was selected RAMPTH Function it is pattern of function applied to the structure. This is the initial condition use when Lacco Time History Non linear analysis is performed .Here Modal damping is modified for 1st three modes. Only difference in Modal load case we add Link so that it specify the results for the isolator. Isolator is an Link/Support element. 16. Run Analysis is performed to interpreted the results

23

4 Load Model

Roof

140 psf

Figure 8: Model with dead load (We can see on left hand side color band Load applied to the Floors 140 psf)

24

125 psf

50 psf

Figure 9: Model with Live Load (We can see on left hand side color band Load applied to the Floors 125psf And 50 psf to the roof)

25

5 Analysis Response Spectrum Analysis Results: To perform analysis for Seattle Spectra generated using IBC 2006 by SAP. Model1, Model2, Model3 have been tested using spectra and results are obtained. 1. Damping: In all three model damping ratio was assigned to 0.05 during an I/P of generation of spectra, No advance damping was defined for the model. 2. Accelerations: For each mode acceleration are printed in local co-ordinate system, so when we proceed for reading results in this project it identified by the symbol U1 Acc And U2 Acc.(this value are the acceleration for each mode are the actual values interpolated at the modal period from the spectra curve.) 3. Modal Amplitude: The response spectrum modal amplitude give the multipliers of the mode shapes that contribute to the displaced shape for the each direction of acceleration load. In the result it is identified as U1AMP & U2AMP. 4. Displacement: Noted the Joint displacement at point A,B & C for each floor ( Refer the plan drawing) for the Model1, Model2 & Model3 under the application of seismic spectra. In the result it is identified as U1 & U2 5. Shear Force and Bending Moment: For the Model1, Model2 & Model3 forces and moment were noted under a “Seattle spectra” 6. Base reaction: For the models base reaction are noted , which says the total forces and moment about the global origin required of the supports (restraint and spring) to resist the inertia forces due to response spectrum loading. In the result they are identified as as in the gloabal co-ordinate Fx, Fy, Mx & My) Modal Analysis Results: To perform analysis for Acclerated load applied in Ux and Uy and look for Modal participation mass ratio. The idea behind the modal analysis is to decouple vector 1. Period (T) in sec which identified in the results which represent the period of a mode for complete system. 2. Eigen value is obtained for each mode Identified in the results as ω2 in rads/sec 3. Modal Mass was seen in the result as an unity.. 4. Modal Stiffness was seen as modal eigenvalue. 5. Modal Load applied in Ux and Uy there dynamic participation was checked. 6. Modal Participating Mass ratios were checked that it reaches to 99% of Cumulative sums of participating mass ratio for all modes). In the result it is identified Sum of Ux and Sum of Uy.

26

Model 1 3-D Four Story building without shear wall. And performed Response Spectrum Analysis for the model.

U1 1.752”

Figure10: Showing displacement under Seattle Spectra-Model 1

Maximum value Shear force was noticed at base level Int Col 1&2 Vu Dynamic due to Seattle Spectra Should be considered for the design

Figure11: Showing shear force under Seattle Spectra- Model 1

27

Figure12: Showing bending moment (at grid line 2) under Seattle SpectraModel1 `Area of interest strong columns needed (Playing with reinforcement criterion good idea to see the change in behavior)

28

Model 2 3-d four Story building with shear wall. And performed Response Spectrum Analysis for the model.

U2 1.19

Figure13: Showing Displacement under Seattle Spectra-Model 2

Maximum value Shear force was noticed at top level Int Col 3&4 V Dynamic due to Seattle Spectra Should be considered for the design

Figure14: Showing shear force under Seattle Spectra- Model 2

29

Maximum Moment in beam was noticed adjacent to the wall

Figure15: Showing bending moment (at grid line 1) under Seattle Spectra-Model2

Model 3 3-d four Story (soft story, 1.e. floor ht. variation was performed) building with shear wall. And performed Response Spectrum Analysis for the model.

U2 1.138”

Figure 16 Showing displacement under Seattle Spectra-Model 3

30

Maximum Shear force was noticed At Int Col. 3

Figure17 Showing shear force under Seattle Spectra- Model 3

Maximum Moment in beam was noticed adjacent to the wall

Figure18: Showing bending moment (at grid line 1) under Seattle Spectra-Model3

31

Maximum Moment was Noticed in Int .Col.4

Maximum Moment was noticed in Beam at 3rd floor

Figure19: Showing bending moment (at grid line 2) under Seattle Spectra-Model3

Table 8: Comparison Tables and Result Obtained for 3 models TABLE: Response Spectrum Modal Information Model 1 Displacemen Displacemen t t Floo Join r t in U1 in U2

Model 2 Displacemen Displacemen t t

Model 3 Displacemen Displacemen t t

in U1

in U2

in U1

in U2

Nos

Nos

inch

inch

Inch

inch

inch

inch

1

122

0.435

2.8177

0.1305

0.167

0.1075

0.1418

1

127

0.435

2.8177

0.1305

0.167

0.1075

0.1418

1

132

0.435

2.8177

0.1305

0.167

0.1075

0.1418

2

123

0.8847

5.6134

0.3869

0.466

0.3197

0.3926

2

128

0.8847

5.6134

0.3869

0.466

0.3197

0.3926

2

133

0.8847

5.6134

0.3869

0.466

0.3197

0.3926

3

124

1.4362

8.0379

0.7005

0.8264

0.591

0.6992

32 Floo r

Join t

Displacment

Displacment

Displacemen t

Displacemen t

Displacemen t

Displacemen t

in U1

in U2

in U1

in U2

in U1

in U2

Nos

Nos

inch

inch

Inch

inch

inch

inch

4

125

1.7516

9.4159

1.0139

1.1917

0.9912

1.138

4

130

1.7516

9.4159

1.0139

1.1917

0.9912

1.138

4

135

1.7516

9.4159

1.0139

1.1917

0.9912

1.138

Comparision of Displacement for 3 different models seattle spectra x-dirn displacement model1 seattle spectra-y dirn displacement-model 1 seattle spectra x- dirn displacement model2 seattle spectra y dirn displacement model2 seattle spectra x-dirn displacement-soft story seattle spectra Y-dirn displacement-soft story 4.5 4 3.5 3 2.5 Floors 2 1.5 1 0.5 0 0

1

2

3

4 5 6 Displacement in inch

7

8

Figure 20: Comparison of Displacement for 3 different models

9

10

33

Table 9: Comparison Calculation for finding % reduction of displacement due to soft story Comparison Calculation

Comparison of U2 Model 2 &3

% Displacement Reduction in U1 due to soft story

% Displacement Reduction in U2 due to soft story

inch

inch

%

%

1

0.023

0.0252

17.62452107

15.08982036

2

0.0672

0.0734

17.36882915

15.75107296

3

0.1095

0.1272

15.63169165

15.39206196

4

0.0227

0.0537

2.238879574

4.50616766

Floor

Comparison of U1 Model 2 &3

Nos

Table 10: Comparison of base reaction due to all 3 model & % reduction of base reaction due to soft story TABLE: Base Reactions

Comparison table

OutputCase

GlobalFX

GlobalFY

GlobalMX

GlobalMY

Text SEATTLE SPECTRA ANALYSIS Model 1 SEATTLE SPECTRA ANALYSIS Model 2 SEATTLE SPECTRA ANALYSIS Model 3

Kip

Kip

Kip-in

Kip-in

1336.864

2800.527

1126741.653

561191.13

1954.772

6424.492

2865226.938

844083.04

1883.922

6250.226

2720878

794093.6

Difference (Model2 Model 3)

70.85

174.266

144348.938

49989.437

Reduction in (%) base shear for soft story on comparison of model 2

3.624464

2.712526

5.037958288

5.9223364

34

Table 11: Response Spectrum Analysis Model 1 Response Spectrum Analysis Model 1 Mode

Period

CircFreq

Eigenvalue

U1Acc

U2Acc

U1Amp

U2Amp

Unitless

Sec

(ω)rad/sec

rad2/sec2

in/sec2

in/sec2

in

in

1

0.959617

6.5476

42.871

95.026

316.754

3.19E-17

21.44683

2

0.758166

8.2873

68.68

120.959

403.196

2.25E-16

3

0.575054

10.926

119.38

160.556

535.185

3.822136

3.97E-14 -7.00E17

4

0.33584

18.709

350.02

235.447

784.823

2.250856

5

0.273242

22.995

528.77

235.447

784.823

3.98E-18 -2.30E17

6

0.211691

29.681

880.95

235.447

784.823

-0.30296

6.82E-18

7

0.208873

30.081

904.89

235.447

784.823

1.19E-17

0.4261

8

0.169113

37.154

1380.4

235.447

784.823

7.87E-17

9

0.163768

38.366

1472

235.447

784.823

5.65E-18 -1.00E17

0.168688

10

0.135146

46.492

2161.5

235.447

784.823

-0.05127

9.50E-18

System is not Stiff hence higher value is noticed in natural period

4.24E-17

Table 12: Response Spectrum Analysis Model 2 Response Spectrum Analysis Model 2 StepNum

Period

CircFreq Eigenvalue

U1Acc

U2Acc

U1Amp

U2Amp

Unitless

Sec

rad/sec

rad2/sec2

in/sec2

in/sec2

in

in

1 0.353912

17.754

315.19

235.447

784.823 2.109278

2 0.211951

29.645

878.8

235.447

3 0.154698

40.616

1649.6

235.447

784.823 0.011517 2.515816 -1.00E784.823 16 1.50E-15

4 0.088961

70.629

4988.4

235.447

784.823 0.069433

-0.00171

5 0.075264

83.482

6969.2

232.548

775.159

0.00206

-0.00549

6 0.065681

95.663

9151.4

214.929

716.43 0.000214

-0.00599

7 0.063453

99.021

9805.2

210.833

702.777

-0.00042 0.013416

8 0.062264

100.91

10183

208.648

695.494

-0.00076

-0.0018

9 0.057624

109.04

11889

200.118

667.06 0.000308

-0.01643

10 0.054874

114.5

13111

195.062

650.205

-0.00019

-0.09369

-0.07391

System is Stiff hence reduction is noticed in natural period

35

Table 13: Response Spectrum Analysis Model 3 Response Spectrum Analysis Model 3 StepNum

Period

CircFreq Eigenvalue

U1Acc

U2Acc

U1Amp

U2Amp

Unitless

Sec

rad/sec

rad2/sec2

in/sec2

in/sec2

in

in

1

0.346101

18.154

329.58

235.447

784.823

1.96549

-0.08366

2

0.205136

30.629

938.16

235.447

784.823

0.010074 2.313225

3

0.150421

41.771

1744.8

235.447

784.823

9.40E-18

7.71E-15

4

0.093568

67.151

4509.3

235.447

784.823

0.081621

-0.00178

5

0.0735

85.486

7307.8

229.304

764.348

0.000886

-0.00533

6

0.063909

98.315

9665.9

211.671

705.571

-0.00014

0.006343

7

0.062567

100.42

10085

209.205

697.349

0.000228

-0.01629

8

0.060883

103.2

10651

206.108

687.028

0.000352

0.00447

9

0.057078

110.08

12118

199.114

663.713

0.000086

-0.07213

10

0.056205

111.79

12497

197.509

658.364

-0.00026

-0.04667

System is very stiff hence reduction is noticed in natural period

Table 14: Comparison of effect of soft story in RSA Floor

Model 3 (Soft Story)

Displacement Reduction

Displacement Reduction

Base Reactions

No.

Comparison of stiffness increase in stiffness in Model 3

% Displacement Reduction in U1 due to soft story %

% Displacement Reduction in U2 due to soft story %

Reduction in (%) base shear for soft story on comparison of model 2

% 1

29.9879 % increase

17.62452107

15.08982036

2

29.9879 % increase

17.36882915

15.75107296

3

29.9879 % increase

15.63169165

15.39206196

4

48.73702%decrease

2.238879574

4.50616766

(Fx) %

(Fy) %

(Mx) %

(My) %

3.62446362

2.7125258

5.0379583

5.92233641

Interpretation: 1. Discussion about displacement comparison Model 1 with Model 2 and 3 we can see clearly from the graph (fig. 20) where displacements for Model 1 is very high for U1 and U2 . The reason is very simple that the provision of shear wall was made in Model 2 and 3 which was oriented in all direction as can be seen from the model. It provides building with seismic resistance. So provision of shear wall is one of the seismic resistant structures.

36

2. For member force for Model 1 we can see from (fig. 11 & 12) when analyzing for complete building the maximum Shear force and Bending Moment was noticed in the grid line 2 of plan (the strong columns and resistant to seismic is needed at the interior column at first floor.) 3. When comparing the displacement Model 2 with 3 from (fig. 20 and Table 14) we can notice that due to increase of stiffness in floor 1, 2 & 3 there is a reduction in displacement in the floors. But when there is a decrease of stiffness in floor 4 there is a reduction in displacement in the 4th floors but now the reduction of displacement is less compare to the floor 1, 2 & 3.Overall soft story can achieve reduction in displacement if stiffness is rearrange in the building. 4. When comparing the base reaction in global direction for Model 2 with 3, we can notice that reduction of base reaction in Model 3 (Table 14 shows the value). So we can say to resist the inertia forces due to Response spectrum is less for the soft story. 5. Warily studies was performed for the member forces for Model 2 with 3(refer fig. 14, fig 15, fig 17, fig 18 & fig 19) where comment are listed by noticing the Mu Seattle & Vu Seattle. Based on the maximum value and use of some conservative reinforcing pattern should be adopted. For the model I have consider #9 longitudinal bars and #4 Confinement bars and confinement ties (for Beams and Columns) i. The ductile frame joint based on the high seismic study (there are standard guidelines available to adopt in high seismic region) ii. Requirement of the boundary members should be adopted iii. Seismic Hooks, Cross tie and hoops can be provided iv. To design for Frame Flexural Members should be adopted v. Transverse Confinement in the Flexural member should be adopted. vi. Providing a Bond Beam. (Information obtain from Michael R Lindeburg, “Seismic Design of Building Structure”) 6. Study of Modal Analysis for an ndof we have “n” no. of mode for the project we don’t need n no. of modes to evaluate results for all three model for all 3 model the result were obtain for 10 modes modal participating mass ratios reaches to 98% and modal load participation factor reaches 100% of what we applied (i.e. Seattle Spectra) in both U1 and U2 .So result are complete 7. When comparing Model 1,2 and 3 result for each mode shape was check for correctness when we look to structural o/p of the SAP 2000, looking for Modal Participation factor in which I obtained for each mode . i. Modal Mass is an Unity (speaking in terms of theoretical terms Modal mass matrix is an identity matrix) ii. Modal Stiffness for each mode was obtained as (natural frequency )2 equal to eigenvalue which is tabulated in (eigenvalue- table 11,12 &13) 8. Comparing period for all 3 model we can see model 1 has very high period compare to model 2 and 3. When we compare Model 1 period with theoretical period based on IBC

37

2006. We can say theoretical period are very approximate value design cannot be performed based on theoretical basis. Comparing Model 2 with Model 3. Reduction in period is notice for each mode by mode comparison. 9. Modal Amplitude obtain from (table 11,12 &13) identified as U1AMP & U2AMP are the multiplier of the mode shapes that has contribute to displaced shape. We can notice from the table clearly that U2 Amp contribution is very high for Model 1 because it does not have resistive wall in the model. For Comparing Model 2 with Model 3 U2 Amp contribution is high for Model 2 compare to Model 3. 10. Analysis was performed by now changing 30% of IBC 2006 in Y-direction & 100% of IBC 2006 in X-direction. The Displacement result where lower. The control direction of loading is 100% of IBC 2006 in Y-direction and 30% of IBC 2006 in X-directions

Model 4 3-d four Story building with shear wall. And performed Time History Analysis for the model.

Maximum Joint Displacement at Joint 90 U 1 3.557542 inch

Maximum Joint Displacement at Joint 10 U 2 3.2059 inch

Figure 21 Showing displacement under LACCO NOR earthquake record-Model 4

38

Figure 22 Comparison of Joint displacement under LACCO NOR earthquake record-Model 4 (U2 displacement is at 9.62 sec 0.4741 inch)

39

Figure 23 Comparison of Joint displacement under LACCO NOR earthquake record-Model 4 (U1 displacement is at 7.31 sec 0.9826 inch) Table 15: Modal Periods and Frequencies for LACCO NOR earthquake-Model 4 TABLE: Modal Periods And Frequencies StepNum Unitless 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Period Sec 0.353912 0.211951 0.154698 0.088961 0.075264 0.065681 0.063453 0.062264 0.057624 0.054874 0.0511 0.050546 0.044782 0.041648 0.040872

CircFreq rad/sec 17.754 29.645 40.616 70.629 83.482 95.663 99.021 100.91 109.04 114.5 122.96 124.31 140.3 150.86 153.73

Eigenvalue rad2/sec2 315.19 878.8 1649.6 4988.4 6969.2 9151.4 9805.2 10183 11889 13111 15119 15452 19685 22760 23633

Table 16: Base Reaction for LACCO Spectra-Model 5 TABLE: Base Reactions OutputCase Text LACCO SPECTRA

CaseType Text LinModHist

GlobalFX Kip 1802.357

GlobalFY Kip 2499.414

GlobalMX Kip-in 1128363.425

GlobalMY Kip-in 798825.409

40

Model 5 3-d four Story building (here rubber isolator and mass slab is provided) with shear wall. And performed Time History Analysis for the model.

Rubber Isolator Provided

Figure 24 Model 5 Showing Rubber Isolator.

Figure 25 Comparison of Joint displacement under LACCO NOR earthquake record-Model 5 (U2 displacement very little difference between each floor displacement)

41

Figure 26 Comparison of Joint displacement under LACCO NOR earthquake record-Model 5 (U1 displacement is very high)

Figure 27 Comparison of Joint Vs Base Shear under LACCO NOR earthquake record-Model 5 (U1 displacement is very high)

42

Table 17: Modal period and frequencies-Model 5 TABLE: Modal Periods And Frequencies StepNum Unitless 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Natural Period is very high Period Sec 12.20939 12.20847 11.91553 0.332325 0.311778 0.239744 0.168803 0.158349 0.135287 0.132027 0.125196 0.122749 0.121475 0.105582 0.102957

CircFreq rad/sec 0.51462 0.51466 0.52731 18.907 20.153 26.208 37.222 39.679 46.443 47.59 50.187 51.187 51.724 59.51 61.028

Eigenvalue rad2/sec2 0.26483 0.26487 0.27806 357.46 406.13 686.85 1385.5 1574.4 2157 2264.8 2518.7 2620.1 2675.4 3541.5 3724.4

Table 18: Comparison of Period of Model 4 & Model 5 Comparison of Periods

Mode Unitless 1 2 3 4 5 6 7 8 9 10 11

Period Isolator (Tb) Sec 12.20939 12.20847 11.91553 0.332325 0.311778 0.239744 0.168803 0.158349 0.135287 0.132027 0.125196

Period Fixed (Tf) Sec 0.353912 0.211951 0.154698 0.088961 0.075264 0.065681 0.063453 0.062264 0.057624 0.054874 0.0511

(Tb/Tf) 34.49838 57.60045 77.02447 3.735626 4.142459 3.650127 2.660284 2.543187 2.347754 2.406003 2.45002

43

Mode 12 13 14 15

Period Isolator (Tb) 0.122749 0.121475 0.105582 0.102957

Period Fixed (Tf) 0.050546 0.044782 0.041648 0.040872

Figure 28 Layout of Link Element

Figure 29 and Figure 30 shows the plot of Isolator deformation

Figure 29 Isolator Deformations –Model 5-Link Set 1

(Tb/Tf) 2.428461 2.712585 2.535104 2.519011

44

Figure 30 Isolator Deformations –Model 5-Link Set 2 Table 19: Comparison of Base Reaction of Model 4 & Model 5 TABLE: Base Reactions OutputCase Text LAC-Model 4 LAC-Model 5 % Reduction of Base Reaction due to isolator addition

GlobalFX Kip 1802.357 0

GlobalFY Kip 2499.414 106.819

GlobalMX Kip-in 1128363 28316.15

GlobalMY Kip-in 798825.4 1124.768

100

95.72624

97.49051

99.8592

Interpretation: 1. When comparative study done between Model 4 & Model 5 , Base Isolation lengthen the period the fundamental vibration of the structure which can be seen from (table 17 and table 18) and because of isolator provision in Model 5 reduces the pseudo acceleration for the mode. 2. In Model 5 the first vibration mode of isolated structure involves deformation in the isolator link element. The structure is moving as a rigid body on the top of the isolator. 3. From fig. 21 we can see that maximum displacement (U2 & U1) in the structure occurs at different time. From fig. 22 & 23 we can see the difference of displacement in each story. Now at same point we study for model 5 (fig. 25 & 26) we can see there is no difference in the displacement at each level, the effect of isolator is that structure is moving as a rigid body on the top of the isolator. 4. From fig 29 & 30 we can see clearly deformation in the isolator is very high. 5. When comparing the Model 4 & Model 5 for base reaction we can see the inertia force required to resist the structure from LACCO Nor earthquake record is less for Model 5

45

as we can make out from comparison table 19 were we can see that due to provision isolator to the building the reduction of the earthquake forces imparted to the structure. It is no surprise that reduction in base shear is a pink in health for Structure.

46

6 Conclusions 1. After vigilant assessment we can distinguish that for concrete structures, additional development work is required to develop a completely rational method. As we can see that RSA assessment is restricted to linear analysis as RSA analysis have one of the limitation it does not perform nonlinear analysis. When looking to the Model 1, 2 & 3 Model 3 is preferable compare to other 1 & 2, reason is because if we know were to put what size of columns and beams. (if we work out with right Math work for assembling stiffness & rearrangement we can achieve reduction in the displacement due to pseudo ground acceleration) 2. To obtain rational design forces for the concrete member it will be good idea to analyze the structure 3 or 4 earthquake record using time history analysis as they can furnish the design forces required for the critical area. The forces obtain in Model 1, 2,& 3 would be an good approximation for V dynamic & M dynamic but it will be always be good idea to scale out higher value then what we obtain. 3. Time history analysis performed for Model 4 & 5 reduction in base shear was achieved significantly, due to addition of isolator. Hence effectiveness of reduction of earthquake– induced forces in a model 5 was achieved by provision of isolator. 4. If System is very stiff there will be reduction in the natural period, which can be noticed a in the Period comparison for Model 1, 2& 3. For a Model3 it is very stiff system so we can say it is mass sensitive so if we want change in behavior of the system we have to look at the mass and based on that we can achieve the changes in the system (Tuned mass system would be an good recommendation)

47

References

1. Anil K. Chopra, “Dynamics of Structures- Theory and Applications to Earthquake engineering”, Pearson Prentice Hall, NJ, ISBN 0-13-156174 (Obtained from The TISCH Library at Tufts University). 2. Mario Paz, “Structural Dynamics- Theory and Computation”, 5th edition, Kluwer Academic Publisher, Boston, ISBN 1-4020-7667-3 ( Obtained from Lehigh University) 3. Ajaya Kumar Gupta, “Response Spectrum Method – In Seismic Analysis and Design of Structures” CRC Press, Boca Raton, ISBN 0-8493-8628-4 (Obtained from Union College) 4. W.F.Chen & E.M.Lui, “Earthquake Engineering for Structural Design”, CRC Press, Boca Raton, ISBN 0-8493-7234-8 (Obtained from New York State Library) 5. CSI- Introductory Tutorial & Reference Manual for SAP 2000-Linear and Nonlinear Static and Dynamic Analysis and Design of Three- Dimensional Structures, Berkeley CA 6. Michael R. Lindeburg & Majid Baradar, “Seismic Design of Building Structures”, Professional Publications Inc, Belmont, CA, ISBN 1-888577-52-5 (Obtained from Library CECIL C TYRRELL) 7. International building Code 2006- ISBN 1-58001-251-5 (Obtained from Syracuse University-Civil & Environmental Department) 8. Edward L. Wilson, “ Three Dimensional Static and Dynamic Analysis of Structures- A physical approach with Earthquake Engineering” (Obtained from Website) 9. SAP 2000 Software- Syracuse University Civil Engineering Computer Lab.