# Essay: Methodology and problem formulation

3.1 General

In present scenario, we have seen that whenever there is an earthquake a lot of damage occurs. Even if the probability of occurrence of earthquake within the life span of structures is very less, strong ground motion would generally cause greater damage to the structure. Hence, it becomes important that the building must be adequate in resisting earthquake. So building with shear wall helps in resisting lateral load. The Non-linear static analysis of shear wall gives the desired base shear and the story drift which it can encounter during its life.

 Non-linear Static Pushover analysis

Pushover analysis is a term used for the non-linear static analysis of frames. The practical method used for evaluating the displacement, time period etc is most done by pushover analysis. It is the procedure in which the magnitude of the structural loading is incrementally increased in accordance with a certain predefined pattern. With the increase in the magnitude of the loading, weak links and failure modes of the structure are found. The loading is monotonic with the effects of the cyclic behaviour and load reversals being estimated by using a

modified monotonic force-deformation criteria and with damping approximations.

The static pushover analysis is becoming a popular tool for seismic performance evaluation of existing and new structures. The expectation is that the pushover analysis will provide adequate information on seismic demands imposed by the design ground motion on the structural system and its components. Further, Indian buildings built over past two decades are seismically deficient because of lack of awareness regarding seismic behaviour of structures. The widespread damage especially to RC buildings during earthquakes exposed the construction practices being adopted around the world, and generated a great demand for seismic evaluation and retrofitting of existing building stocks.

The pushover analysis of a structure is a static non-linear analysis under permanent vertical loads and gradually increasing lateral loads. The equivalent static lateral loads approximately represent earthquake induced forces. A plot of the total base shear versus top displacement in a structure is obtained by this analysis that would indicate any premature failure or weakness. The analysis is carried out up to failure, thus it enables determination of collapse load and ductility capacity.

 Purpose of Pushover analysis

The purpose of pushover analysis is to evaluate the expected performance of structural systems by estimating performance of a structural system by estimating its strength and deformation demands in design earthquakes by means of static

inelastic analysis, and comparing these demands to available capacities at the performance levels of interest.

The evaluation is based on an assessment of important performance parameters, including global drift; inter story drift, inelastic element deformations (either absolute or normalized with respect to a yield value), deformations between elements, and element connection forces (for elements and connections that cannot sustain inelastic deformations).

The inelastic static pushover analysis can be viewed as a method for predicting seismic force and deformation demands, which accounts in an approximate manner for the redistribution of internal forces that no longer can be resisted within the elastic range of structural behavior.

3.2 Problem Formulation

Different shear wall models having edge column and without edge column are consider with different thickness and story height. The models which are having edge column are confined with high reinforcement whereas those without edge column are unconfined.

 Materials properties

Concrete: The concrete has a uniaxial compressive strength fc’. Under uniaxial compression, the concrete strain εo corresponding to the peak stress fc’ is usually around the range of 0.002–0.003 and used in the analysis is εo 0.003. The Poisson’s ratio νc of concrete under uniaxial compressive stress ranges from

about 0.15-0.22, with a representative value of 0.19 or 0.20. In this study, the Poisson’s ratio of concrete is assumed to be νc 0.2. The uniaxial tensile strength fc’ of concrete is difficult to measure. For this study the value is taken as

ft’= 0.25 √ fc’ MPa

The initial modulus of elasticity of concrete E’is highly correlated to its compressive strength and can be calculated

Ec= 5000 √ fc’ MPa

Table 3.1: Properties of concrete

S.No.

Notations

Value

1.

M-25

25 N/mm2

2.

Ec

25000 N/mm2

3.

fc’

25N/mm2

4.

ft’

1.25 N/mm2

5.

νc

.3

6.

εcrushing

.004

7.

εu

.001

Steel: The elastic modulus, Es, and yield stress, fy, are taken and these values are used in the model. A Poisson’s ratio of 0.3 is used for the steel.

Table 3.2: Steel Properties

Fe-415

415 N/mm2

Es

210000 N/mm2

νs

.3

 Location of Shear Wall:

Shear walls are located on the exterior central portion in all direction. The location is so selected so as to give symmetry to the building. The location of shear wall affects the performance of the building.

The dead load is calculated according the self-weight of the slab and is distributed uniformly. The unit weight of the slab is considered as 25 KN/m3. The live load is assumed to be uniformly distributed on slab and is taken as 2 KN/m2. These loads are applied in vertical direction. The live load is assumed to be taken as 25 % of the total load.

Example Problem:

Table 3.3: 4 Story shear wall with edge column

S. No.

hs (m)

tw

(mm)

Lw

(m)

hw

(m)

Percentage of reinforcement in edge column

Ac

(mm2)

ρ

As

(mm2)

SWE1*

3.5

150

4.0

14

1 %

785

.175

1050

SWE2

3.5

150

4.0

14

2%

1582

.175

1050

SWE3

3.5

200

4.0

14

1%

1099

.131

1048

SWE4

3.5

200

4.0

14

2%

2034

.131

1048

SWE5

3.5

250

4.0

14

1%

1356

.105

1050

SWE6

3.5

250

4.0

14

2%

2814

.105

1050

* SWE stands for Shear wall with edge column

Table 3.4: 6 Story shear wall with edge column

S. No.

hs (m)

tw

(mm)

Lw

(m)

hw

(m)

Percentage of reinforcement in edge column

Ac

(mm2)

ρ

As

(mm2)

SWE7

3.5

150

6.0

21

1 %

785

.308

2772

SWE8

3.5

150

6.0

21

2 %

1582

.308

2772

SWE9

3.5

200

6.0

21

1 %

1099

.224

2688

SWE10

3.5

200

6.0

21

2%

2034

.224

2688

SWE11

3.5

250

6.0

21

1%

1356

.165

2475

SWE12

3.5

250

6.0

21

2%

2814

.165

2475

SWE13

3.5

300

6.0

21

1%

1582

.131

2358

SWE14

3.5

300

6.0

21

2%

3216

.131

2358

Table 3.5: 8 Story shear wall with edge column

S. No.

hs (m)

tw

(mm)

Lw

(m)

hw

(m)

Percentage of reinforcement in edge column

Ac

(mm2)

ρ

As

(mm2)

SWE15

3.5

150

8.0

28

1 %

785

.524

6288

SWE16

3.5

150

8.0

28

2 %

1582

.524

6288

SWE17

3.5

200

8.0

28

1 %

1099

.343

6288

SWE18

3.5

200

8.0

28

2%

2034

.343

6288

SWE19

3.5

250

8.0

28

1%

1356

.302

6040

SWE20

3.5

250

8.0

28

2%

2814

.302

6040

SWE21

3.5

300

8.0

28

1%

1582

.251

6024

SWE22

3.5

300

8.0

28

2%

3216

.251

6024

Table 3.6: 8 Story shear wall without edge column

S. No.

hs

(m)

tw

(mm)

Lw

(m)

hw

(m)

ρ

As

(mm2)

SWN1*

3.5

150

8.0

28

.524

6288

SWN3

3.5

200

8.0

28

.343

6288

SWN5

3.5

250

8.0

28

.302

6040

SWN7

3.5

300

8.0

28

.251

6024

*SWN stands for Shear wall without edge column

Table 3.7: 6 Story shear wall without edge column

S. No.

hs

(m)

tw

(mm)

Lw

(m)

hw

(m)

ρ

As

(mm2)

SWN9

3.5

150

6.0

21

.308

2772

SWN11

3.5

200

6.0

21

.224

2688

SWN13

3.5

250

6.0

21

.165

2475

SWN15

3.5

300

6.0

21

.131

2358

Table 3.8: 4 Story shear wall without edge column

S. No.

hs

(m)

tw

(mm)

Lw

(m)

hw

(m)

ρ

As

(mm2)

SWN17

3.5

150

4.0

14

.175

1050

SWN19

3.5

200

4.0

14

.131

1048

SWN21

3.5

250

4.0

14

.105

1050

 Non-Linear Static Analysis using SAP 2000

Static pushover analysis is an attempt by the structural engineering profession to evaluate the real strength of the structure and it promises to be a useful and effective tool for performance based design. The ATC-40 and FEMA-273 documents have developed modelling procedures, acceptance criteria and analysis procedures for pushover analysis. These documents define force deformation criteria for hinges used in pushover analysis.

The SAP2000 static pushover analysis capabilities, which are fully integrated into the program, allow quick and easy implementation of the pushover procedures prescribed in the ATC-40 and FEMA-273 documents for both two and three-dimensional buildings.

3.3 Modeling in SAP 2000

Modeling and analysis is done using the SAP 2000, 16 models of shear wall with and without edge column are generated and there pushover curves are obtained. Models with different thickness and reinforcing ratio are taken for parametric study and to obtain the base shear and displacement for different story height.

 Procedure

I. The basic computer model (without the pushover data) in the usual manner using the graphical interface of SAP2000 is created.

II. Defining the material properties and assigning the concrete as confined and unconfined concrete.

III. The confined and unconfined concrete are now assigned with the reinforcement in longitudinal and transverse direction.

IV. After that, the sections are assigned with confined and unconfined concrete.

VI. The model is run for the Non-linear static pushover analysis. And the pushover curve is obtained

4

RESULTS

—————————————————————————————————

The Non-linear static analysis of shear wall results is obtained in the form of base shear and displacement curves. The different pushover curves for different story height of shear wall is obtained that are drawn for different thickness. It is to be noted that the results obtained are within the permissible limits and as those given in different codes.

The results are displayed in the form of curve. The curves are divided into three sections that is in the first section the shear wall with edge column for different story i.e. 4, 6 and 8 story is drawn for different thickness having the percentage of reinforcement varying as 1% and 2%. In the second section the shear wall without edge column is drawn for different height. Finally the base shear obtained is compared with the design value.

Hence, from the results it is tried to give an idea to select the required thickness that is suitable for a particular story height of the building.

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60

BASE SHEAR (KN)

DISPLACEMENT (mm)

BASE SHEAR VS DISPLACEMENT

150 MM

200 MM

250 MM

0

200

400

600

800

1,000

1,200

0 10 20 30 40 50 60

BASE SHEAR (KN)

DISPLACEMENT

BASE SHEAR VS DISPLACEMENT

150 MM

200 MM

250 MM

Section 1: Pushover curves for different thickness

Fig. 4.1 Base shear and displacement curve for 4 story shear wall with edge column having 1% reinforcement

Fig. 4.2 Base shear and displacement curve for 4 story shear wall with edge column having 2% reinforcement

 From the above figures it is evident, as the thickness increases the base shear increases but after a particular point it varies constantly.

Fig. 4.3 Base shear and displacement curve for 6 story shear wall with edge column having 1% reinforcement

Fig. 4.4 Base shear and displacement curve for 6 story shear wall with edge column having 2% reinforcement

0

200

400

600

800

1000

1200

1400

0

20

40

60

80

100

BASE SHEAR (KN)

DISPLACEMENT (mm)

BASE SHEAR VS DISPLACEMENT

150 MM

200 MM

250 MM

300 MM

0

200

400

600

800

1000

1200

1400

1600

1800

0

20

40

60

80

100

BASE SHEAR (KN)

DISPLACEMENT (mm)

BASE SHEAR VS DISPLACEMENT

150 MM

200 MM

250 MM

300 MM

Fig. 4.5 Base shear and displacement curve for 8 story shear wall with edge column having 1% reinforcement

Fig. 4.6 Base shear and displacement curve for 8 story shear wall with edge column having 2% reinforcement

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

20

40

60

80

100

120

BASE SHEAR (KN)

DISPLACEMENT (mm)

BASE SHEAR VS DISPLACEMENT

150 MM

200 MM

250 MM

300 MM

0

500

1000

1500

2000

2500

0

20

40

60

80

100

120

BASE SHEAR (KN)

DISPLACEMENT (mm)

BASE SHEAR VS DISPLACEMENT

150 MM

200 MM

250 MM

300 MM

Fig. 4.7 Base shear and displacement curve for 4 story shear wall without edge column

Fig. 4.8 Base shear and displacement curve for 6 story shear wall without edge column

0

100

200

300

400

500

600

700

0

10

20

30

40

50

60

BASE SHEAR (KN)

DISPLACEMENT (mm)

BASE SHEAR VS DISPLACEMENT

150 MM

200 MM

250 MM

0

200

400

600

800

1000

1200

1400

0

20

40

60

80

100

BASE SHEAR (KN)

DISPLACEMENT (mm)

BASE SHEAR VS DISPLACEMENT

150 MM

200 MM

250 MM

300 MM

Fig. 4.9 Base shear and displacement curve for 8 story shear wall without edge column

 The pushover curve for shear wall without edge column shows that the base shear increase linearly up to reaching its peak value then, there is fall in the base shear as the thickness increases. As we can see that after that the base shear remains constant for different thickness.

0

200

400

600

800

1000

1200

1400

1600

1800

0

20

40

60

80

100

120

BASE SHEAR (KN)

DISPLACEMENT (mm)

BASE SHEAR VS DISPLACEMENT

150 MM

200 MM

250 MM

300 MM

Section 2: Base Shear comparison with design value for edge column

Fig. 4.10 Base shear Comparison with Design Value for 4 story shear wall

Fig. 4.11 Base shear Comparison with Design Value for 6 story shear wall

 From figure 4.10, it is clear that for 4 story shear with edge column 150 mm thickness can be used. And from figure 4.11, i.e. for 6 story height the thickness required is 200 mm ,250 mm thickness can also be used.

0

200

400

600

800

1000

1200

1400

150

200

250

BASE SHEAR (KN)

THICKNESS (mm)

With Edge column

Design Value

0

500

1000

1500

2000

2500

150

200

250

300

BASE SHEAR (KN)

THICKNESS (mm)

With Edge Column

Design value

Fig. 4.12 Base shear Comparison with Design Value for 8 story shear wall for 1 % reinforcement

Fig. 4.13 Base shear Comparison with Design Value for 8 story shear wall for 2 % reinforcement

 For 8 story height the thickness requirement 250mm and 300 mm as shown in above figures.

0

500

1000

1500

2000

2500

3000

3500

4000

150

200

250

300

BASE SHEAR (KN)

THICKNESS (mm)

With Edge Column

Design Value

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

150

200

250

300

BASE SHEAR (KN)

THICKNESS (mm)

With Edge Column

Design Value

Section 3: Base Shear comparison with design value without edge column

Fig. 4.14 Base shear Comparison with Design Value for 4 story shear wall without edge column

Fig. 4.15 Base shear Comparison with Design Value for 6 story shear wall without edge column

0

200

400

600

800

1000

1200

1400

150

200

250

BASE SHEAR (KN)

THICKNESS (mm)

Without Edge Column

Design Value

0

500

1000

1500

2000

2500

150

200

250

300

BASE SHEAR (KN)

THICKNESS (mm)

Without Edge Column

Design Value

Fig. 4.16 Base shear Comparison with Design Value for 8 story shear wall without edge column

 From figure 4.14, we can see that the base shear value without edge column and design values are approximately equal for 150 mm thickness. It means that if shear wall without edge column is to be used for 4 story then, 150 mm thickness is sufficient. Similarly, we can say for 6 story and 8 story the thickness requirement are 250 mm and 300 mm respectively.

0

500

1000

1500

2000

2500

3000

3500

4000

150

200

250

300

BASE SHEAR KN)

THICKNESS (mm)

Without Edge Column

Design Value

5

Conclusion and FUTURE SCOPE

——————————————————————————————————-

5.1 Conclusion

It is evident that with the use of shear wall lateral stiffness of the structure increases. The building having shear wall performs better in earthquake. The displacement also reduces with the use of shear wall. Hence, it becomes very important to design the shear wall correctly and its location so as to fully utilized it to resist the lateral forces.

Now a days multi storied building are constructed using shear wall so its proper analysis and design must be done so that structure performance increases.

Provision of shear wall results in a huge decrease in base shear and roof displacement both symmetrical building and un-symmetrical building. Pushover curves show non-ductile behavior of the building, because almost all the seismic load is carried by the shear walls and at very small displacement, hinges start forming in shear walls. This indicates that strengthening of the shear walls in the building is required.

The present study done on shear wall with and without edge column concludes that:

 When the thickness of the shear wall increases the base shear increases but the displacement decreases.

 For lower height of building, minimum thickness can be taken as 150mm as we have seen in 4 story shear wall thickness less than 150 mm can be taken and higher thickness will not be economical.

 While changing the percentage of reinforcement in edge column, it is clear that as the percentage increases the base shear obtained matches with design value.

 There is a need of modification in percentage of reinforcement in edge column as given in codes.

5.2 Future Scope:

It is the part of research to analyse the shear wall with different width and thickness of the edge column. Since the whole study is based on uniform thickness of wall it would be part of research to take different thickness of wall section. Also, work can be done on location of shear wall, number of shear wall.