# Influence of P-Δ Effects on the Seismic Fragility of RC Precast Structures

**Info:** 11144 words (45 pages) Dissertation

**Published:** 9th Dec 2019

**Tagged:**
Geology

**Table of Contents**

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**Abstract**

- Introduction and methodology
- General
- Precast Concrete
- P-Δ Effect
- Single-storey and Multi-storey Precast Frame
- Literature Review
- Effects of P-Δ
- Seismic Performance of Precast Structures
- Performance of single-storey building and multi-storey building

- Methodology

- Theoretical and research challenge

2.1 General

2.2 OpenSees

2.3 Modelling a structure in OpenSees

2.4 Dynamic analysis of OpenSees Advanced Examples Manual

- Test results

3.1 General

3.2 Case 1

3.2.1 Considering Reinforcement Ratio, ρ = 1%

3.2.1.1 Linear Coordinate Transformation

3.2.1.2 P-Δ Coordinate Transformation

3.2.2 Considering Reinforcement Ratio, ρ = 2%

3.2.2.1 Linear Coordinate Transformation

3.2.2.2 P-Δ Coordinate Transformation

3.3 Case 2

3.3.1 Considering Reinforcement Ratio, ρ = 1%

3.3.1.1 Linear Coordinate Transformation

3.3.1.2 P-Δ Coordinate Transformation

3.3.2 Considering Reinforcement Ratio, ρ = 2%

3.3.2.1 Linear Coordinate Transformation

3.3.2.2 P-Δ Coordinate Transformation

- Analysis and discussion

4.1 General

4.2 Analysis and discussion of test result

- Conclusion

5.1 General

5.2 Conclusion

**References**

**Abstract**

It has been seen that precast structures is emerging as a potential structure in the reinforced concrete construction sector, in many European countries. It is due to the fact that it offer many advantages as compared to in-situ concrete. Furthermore, precast concrete is more preferred as compared to in-situ because it is economical, less time consuming and requires less amount of labour. In the present study, multi-storey precast column is being analysed and stimulated. As it has been seen that single-storey precast concrete has been ensuring a safer and economic structure. However, many on-going research on multi-storey structure have proven to be safe and economic. In this study two different dimensions of non-linear precast concrete column have been designed as per Eurocode-8 for the analysis of P-Δ effect or geometric nonlinearities. The analysing of seismic response will be carried out by OpenSees software. Furthermore, to study the seismic response from the results, data’s are being graphically represented with the help of Microsoft excel.

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**CHAPTER 1**

INTRODUCTION AND METHODOLOGY

**1.1 General**

This chapter will explain about precast structures and benefits of precast structures. It will also explain about P-Δ effect. Effects of P-Δ effects and seismic performance of precast structures is also explained.

**1.2 Precast Concrete**

Precast concrete is a factory manufactured construction product, which is produced by casting concrete in moulds and after curing it is transported to the site, whether it is prestressed or reinforced with conventional reinforcement bar. The difference between cast-in-situ and precast is that in cast-in-situ moulds are assembled at the construction site and ready mix concrete trucks deliver the wet concrete and once the concrete is cured the moulds are removed. Since both are working with the same material, there are bound to be some similarities in the processes as both require formwork to hold the wet concrete in the desired shape until it is cured and both have some type of steel reinforcement to help concrete meet the design requirements. Most common precast structures system used in Europe are Dowel joints or hinge connection system between beams and columns with ties. Even prestressed concrete is not much different from precast concrete, however in prestressed concrete the structure is being placed under compression prior to supporting any loads beyond its own dead weight.

Figure 1.1 Precast Concrete ( Source- http://www.jp-uk.com)

Some of the advantages and disadvantages of precast concrete are as follows

Advantages:

• Reduce construction time and cost.

• Manufactured in control environment condition.

• Better quality of structural element.

• High durability and load capacity

Disadvantages:

• Difficulties in handling and transportation.

• Smooth Connection.

• High Initial capital cost.

Depending upon the load bearing capacity Precast concrete are divided into following structural typology:

- Large-panel systems
- Frame systems
- Slab-column systems with walls
- Mixed systems

Connection of precast concrete elements

Precast element is either connected or assembled by dowel joints and later on it is grouted. Grout is a mixture of cement ,water and sand .

Figure 1.2 Typical column to column connection (Source- http://paradigm.in)

Figure 1.3 Typical column to beam connection (Source- http://paradigm.in)

Figure 1.4 Typical slab to beam connection (Source- http://paradigm.in)

**1.3 P-Δ Effect**

During an earthquake action, a structure is subjected to an horizontal displacement , which is generally referred as P-Δ effects. Therefore, P-Δ effects or geometric nonlinearities can be defined as the displacement amplification caused by the gravity loads on the deformed structure by any horizontal action. P-Δ is mainly governed by two important factor- the magnitude of the gravity loads and horizontal displacement influenced by earthquake actions. Thus during design process these factors needs to considered very carefully. However, structural instability can be overcome by increasing member size thus increase of structural stiffness. This type of problem can be seen in single-story precast concrete as columns are connected by pinned connections. According to Eurocode-8 [2], second-order effects (P-Δ effects) need not be taken into account if the following condition is fulfilled in all story

θ=Ptot. dr/(Vtot.h))≤0.1

Where

θ

– Inter storey drift sensitivity coefficient

P_{tot} – Total gravity load at and above the storey considered in the seismic design situation

d_{r} – Inter storey drift, evaluated as the difference of the average lateral displacen1ents ds at the top and bottom of the story under consideration and calculated in accordance.

V_{tot} – Total Seismic storey shear.

h – Inter storey height.

Figure 1.5 P-Δ about Column (Source- https://wiki.csiamerica.com)

**1.4 Single-storey and Multi-storey**** Precast Frame**

Precast concrete frame structure consists of basic type of structural system (beams, column, roof, floors and walls). These structural system are combined in different ways to fulfil the requirements to obtain an effective structural system. Most common system are beam and column system, floor and roof system, bearing wall system and facade system. [15]

In precast concrete building columns are usually slender columns, connected with socket foundation at the base and column-beam by pinned connection [10]. In multi-storey structure, during inelastic response shear force in the column is higher than the equivalent elastic response and very high seismic magnification were observed et.al. Even capacity design for shear is not straightforward in multi -storey frame system as the distribution of the moment over the height of the column is unknown and changes during the response, whereas capacity design for shear is straight-forward in single storey system [9]

**1.5 Literature Review**

**1.5.1 Effects of P-Δ **

In past, many research had been carried out to investigate the effects of P-Δ effect in the single and multi-storey building. Firstly, researchers defined by neglecting P-Δ effects and later considering by structural coefficient such as lateral stiffness of the structure, ductility demand and axial loads.

Bernal D [7] and MacRae [8] considered P-Δ effects by expanding the quality of the structure keeping in mind the end goal to have a similar ductility demand as of structural response without P-Δ effect. Bernal [7] defined the amplification factor as the proportion of the required strength of SDOF system to achieve a peak displacement ductility demand, with or without P-Δ effects. MacRae [8] proposed that hardening ratio is the most important parameter for P-Δ effects as it controls inelastic deformation and dynamic instability due to geometric nonlinearity.

Ercolino et.al [10], explain about the stability factor (θ), as the ratio between the axial load and lateral stiffness and the structure height. E.F. Black [12], introduced two different stability coefficients which can be used to identify the P-Δ effect during a lateral displacement. First coefficient, which was applied during elastic lateral displacement, produced by an analytical process. It is a function of elastic fundamental period and modal shape of the structure and therefore named as modal-elastic stability coefficient (θme). The second coefficient was produced by reverse analysis process. This coefficient holds a strong correlation with elastic fundamental period and modal shape of the structure and thereafter named as modal-inelastic stability coefficient (θmi).

Hyo-Gyoung Kwak and Jin-Kook Kim [11], investigated the P-Δ effect in slender reinforced concrete (RC) columns for sixty sets of horizontal and vertical earthquake loads and found out that effect of axial force reduces at the end of the column with negligible slenderness ratios.

Matej Fischinger et.al [9], investigated seismic shear magnification in columns of multi-storeys much higher than the predicted elastic procedures by designers. Therefore it was concluded that brittle failure in the column should be avoided and this phenomenon should be noted for the capacity design of cantilevers columns in multi-storey structures. Reasons for shear magnification are Over-strength, Period shift and amplified influence of higher modes.

**1.5.2 Seismic Performance of Precast Structures**

There are different other research work on precast buildings, one such is Ercolino et.al [6], which investigated the vulnerability of the structures with respect to collapse limit state by considering two different seismic codes i.e. Italian Code and EuroCodes. Throughout the paper, comparison between Italian code and Euro code is provided and it was concluded that structures are safe against collapse due to structural over strength with respect to seismic actions.

Seong-Hoon Jeong and Amr S. Elnashai [5], derived a method for seismic fragility for 3D-structures with an irregular plan. It was investigated that proposed method provides more realistic results, by comparing derived fragility curves existing indices. Therefore it was concluded that building should be analysed with significant torsional and bi-directional responses.

As Marianna Ercolino et.al [10], Amplification of seismic effects is important even if θ<0.1 and if P-Δ effects are neglected, structural safety cannot be sufficient.

Thereafter, E.F. Black [12], concluded that both stability coefficients should be applied when the lateral displacement is less than or equal to yield displacement.

**1.5.3 Performance of single-storey building and multi-storey building**

Anze Babic and Matjaz Dolsek [4], conducted dynamic analyses on 12 industrial precast building and suggested that seismic fragility function are not precise to estimate the damage caused in a single structure. Therefore, to get a precise estimation, it would not only require precise data regarding material, it will also require geometry and data about ground motions.

Further, C. Casotto et.al [3], presented a seismic fragility model for Italian Reinforcement Concrete (RC) by random sampling of structures and comparing maximum demand with limit state capacity. From this study, it was indicated that higher fragility is obtained while considering all three components of earthquake action.

After investigating, all these papers it was concluded that shear forces in multi-storey precast structures due to seismic is more than shear forces in a single-storey building.

**1.6 Methodology**

- Learning OpenSees software.
- Practicing non-linear models examples from OpenSees advanced example manual.
- Analysing and stimulating, Nonlinear Cantilever Column: Uniaxial Inelastic Section example from OpenSees.
- Analysing and stimulating, Nonlinear Cantilever Column: Inelastic Uniaxial Materials in Fiber Section example from OpenSees.
- Analysing both these examples for Linear and P-Δ Geometric Transformation.
- Converting the units from psi to SI Unit, and analysing them.
- Designing two different dimension of column according to Eurocode-2 for two storey structure.
- Simulating the seismic response of the column and getting the results by OpenSees.
- Drawing graph of Axial force (Fx) vs Displacement in x-direction (Dx) and Bending Moment (My) vs Curvature (Φy), with the help of Microsoft Excel.
- Discussing and comparing the results.

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**CHAPTER 2**

THEORETICAL AND RESEARCH CHALLENGE

**2.1 General**

The main objective of this project is to investigate seismic response of a two-storey precast non-linear cantilever column by OpenSees software. Further, the result are being graphically represented by Microsoft Excel. The research challenge of this project is that, in this project a two-storey non-linear cantilever column is designed by coding in OpenSees software and since the building is designed as per European stands, parameters are in SI Unit.

**2.2 OpenSees **

This software was created in 1998 by Pacific Earthquake Engineering Research Center (PEER). Network for Earthquake Engineering Simulation (NEES) supported from 2005 to 2014. The key person responsible for developing this software was Gregory L. Fenves. The main purpose of development of this software was for simulation, in order to use modern software techniques, such that it can create a new extensible and open-source finite element software platform for earthquake engineering. The software would encompass both structural and geotechnical engineering. For PEER researchers, the software provided a common research platform within PEER, to educate students and share their work among themselves. Later on this software had been introduced to industries for testing and implementation via PEER. The software that Gregory L. Fenves proposed came to be known as OpenSees. [16]

Figure 2.1 Gregory L. Fenves (President of the University of Texas at Austin) (Source- https://en.wikipedia.org/

wiki/ Gregory_L._Fenves)

OpenSees Stands for OPEN System for Earthquake Engineering Simulation. It is an open-source software framework which is used to create finite element application for structural and geotechnical engineering. Mostly it is being used by earthquake engineering, but it is also used by engineers to study wind, fire and wave effects on structures. OpenSees is primarily written in C++, C and Fortran. The language that binds all these language is C++. OpenSees currently have 160 different element types, 220 material types, 15 solution algorithms, 40 integration strategies and 30 solver types. OpenSees application are used for structural analysis of buildings and bridges, the geotechnical analysis of soil and sometimes combine models consisting of structures and soil models. [16]

As suggested by many researches, OpenSees give similar results as compared other commercial software, until and unless model and analysis are same. Unlike most commercial software which have a graphical interface, where we build the model by mouse click, but OpenSees is an interpreter. Since OpenSees is an interpreter, the input files are written in programming language called as tcl as shown in figure 2.2. Tcl is a scripting language that has been extended for finite element analysis. The extension is done at very low level, however it allows to many great things. [16]

Figure 2.2 Typical Tcl script (Source- http://opensees.berkeley.edu/wiki/index.php/Examples_Manual)

In OpenSees models are represented either as elastic elements or inelastic elements. In elastic element model, elastic beam-column element object is constructed. The construction of elastic beam-column element depends upon the dimension of the structure. However, in inelastic elements force beam-column element object is constructed, which depends on the iterative force-based formulation. Inelastic have two type of section, Uniaxial section and Fiber Section. In uniaxial section, inelastic, uncoupled, axial and flexural stiffnesses are defined. Uniaxial section command is used to construct uniaxial section object which define uniaxial material to present a force-deformation response. In fiber Section, the section is broken down into small fibers in which uniaxial materials are defined independently.

In OpenSees lateral loads are applied in two different types, Static Pushover and Time-Dependent Dynamics Loads. In static pushover, at the highest point control node is located. In static pushover, load is distributed corresponding to mass of the structure along the height of the building. Static analysis is of two types monotonic pushover and reversed cyclic pushover (Figure 2.3). In monotonic pushover, displacement is applied in one direction. However in reversed cyclic pushover, displacement are imposed in positive and negative direction.

Figure 2.3 Types of static pushover (Source- http://opensees.berkeley.edu)

In time-dependent dynamic analysis, acceleration are input in the nodes. It is of five types Uniform Sine-Wave, Multiple-Support Sine-Wave, Uniform Earthquake, Multiple-Support Earthquake and Bidirectional Earthquake. In Uniform Sine-Wave, acceleration are input at all nodes but controlled in one direction. In Multiple-Support Sine-Wave, different displacements are applied at one node in one directions. However in uniform earthquake, earthquake acceleration are input from a file. In this acceleration are applied at all nodes but controlled in one direction. Similarly, in multiple-support earthquake, earthquake displacement are input from a file, but different displacements are applied at a particular nodes in one direction. Further, in bidirectional earthquake, different inputs are applied in two directions, but acceleration are input at all nodes but controlled in one direction. (Figure 2.4)

Figure 2.4 Types of time-dependent dynamic analyse (Source- http://opensees.berkeley.edu)

**2.3 Modelling a structure in OpenSees**

To build a model in OpenSees, every script follows the process of building the model, define and apply gravity loads and define and apply lateral load.

In building the model following steps are followed:

- Model dimensions and degrees of freedom is assigned as required.
- Coordinates of nodes are assigned.
- Boundary Conditions of the node is assigned.
- Mass of the structure is assigned.
- Material properties and connectivity of the element is defined.
- Recorders are defined for output.

After modelling and assigning the material properties, following steps are followed to apply gravity load:

- Nodal or element load is defined.
- Static-analysis parameters are assigned.
- Time is reset to zero

Applying lateral load

- Load pattern are assigned as nodal loads for static analysis and acceleration of ground motion for earthquake. Thereafter, it is analysed. [16]

In this project two different geometric transformation is analysed, i.e. linear transformation and P-Δ transformation. In linear transformation, a linear coordinate transformation object is constructed, to perform a linear geometric transformation of beam stiffness and resisting force from the basic system to the global-coordinate system.

In P-Δ Transformation, a P-Δ Coordinate Transformation object is constructed, which performs a linear geometric transformation of beam stiffness and resisting force from the basic system to the global coordinate system, considering second-order P-Delta effects.

**2.4 Dynamic analysis of OpenSees Advanced Examples Manual**

In these examples column length is 432 inch, column depth is 60 inch, column width is 60 inch and weight of structure is 2000 kips. Cover length is considered as 5 inch., area of longitudinal-reinforcement bars is 2.25 inch^{2} and number of longitudinal-reinforcement bars in column is 16. Grade of concrete is M25, steel yield stress (fy) is 66.8 Kips/inch^{2} and modulus of steel (Es) is 29000 Kips/inch^{2}.

**OpenSees Example 2b. Nonlinear Cantilever Column: Uniaxial Inelastic Section**

Following Figure 2.5 and Figure 2.6, represents Axial Force(Fx) vs Displacement in x-direction (Dx) and Bending Moment (My) vs Curvature (Φy) of the column respectively. The maximum Axial Force(Fx) is 9.9×10^{-2} Kips and maximum Displacement in x-direction (Dx) is 1.82×10^{-3} inch. Similarly, maximum Bending Moment (My) is 5.84×10^{1} Kips-inch and maximum Curvature (Φy) is 4.62×10^{-6} inch^{-1}. The maximum drift is 4.20×10^{-6} inch.

Figure 2.5 Axial Force(Fx) vs Displacement in x-direction (Dx) of the column

Figure 2.6 Bending Moment (My) vs Curvature (Φy) of the column.

**OpenSees Example 2c. Nonlinear Cantilever Column: Inelastic Uniaxial Materials in Fiber Section**

Following Figure 2.7 and Figure 2.8, represents Axial Force(Fx) vs Displacement in x-direction (Dx) and Bending Moment (My) vs Curvature (Φy) of the column respectively. The maximum Axial Force(Fx) is 1.55×10^{-1} Kips and maximum Displacement in x-direction (Dx) is 9.29×10^{-4} inch. Similarly, maximum Bending Moment (My) is 6.21×10^{1} Kips-inch and maximum Curvature (Φy) is 3.46×10^{-6} inch^{-1}. The maximum drift is 2.15×10^{-6} inch.

Figure 2.7 Axial Force(Fx) vs Displacement in x-direction (Dx) of the column

Figure 2.8 Bending Moment (My) vs Curvature (Φy) of the column.

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**CHAPTER 3**

TEST RESULTS

**3.1 General**

In the present chapter results of two different dimension of column are designed as per Eurocode-2 [13]. Also, the comparison of Axial force (Fx) vs Displacement in x-direction (Dx) and also Bending Moment (My) vs Curvature (Φy) are shown for reinforcement ratio (ρ) 1% and 2% of both geometric transformation, Linear Transformation as well as P-Δ Transformation .

Figure 3.1 Cross Sectional of a column (Source-https://digitalcanalstructural.com/product/concrete-column-design.html)

**3.2 Case 1**

Width, b = 1 m

Depth, d = 1 m

Height, h = 10 m

Clear Cover = 5 cm

Weigth of Structure = L x B x H x Density of concrete

= 1 m*1 m*10 m*25 KN/m^{3}

= 250 KN

f_{c }= 40 MPa

E_{c }= 4700

√fcMPa

= 29725.41 N/mm^{2}

In UK characteristic yield stress, f_{yk} = 500 MPa

Therefore, design yield stress, f_{yd} =

fykγs

= 500/1.15 = 435 MPa

Modulus of steel may be taken to be, E_{s} = 200 GPa

**3.2.1 Considering Reinforcement Ratio, ρ = 1%**

Reinforcement Ratio, ρ =

AsAc

Where A_{s }= Area of Steel

A_{c} = Area of Concrete (b*d)

Area of Steel A_{s} = 0.01*1*1

= 0.01 m^{2}

Considering 24 mm bar dia

Area of one bar =

πr2

= 3.141*(0.012)^{2} m^{2}

= 0.000452304 m^{2}

No. of Reinforcement =

AsArea of one bar

= 22

**3.2.1.1 Linear Coordinate Transformation**

By considering reinforcement ratio (ρ) 1%, linear coordinate transformation is analysed and stimulated. Figure 3.2 and Figure 3.3 as shown below presents graphical representation of Axial Force (Fx) vs Displacement in x-direction (Dx) of 1^{st} storey and 2^{nd} storey column respectively. The minimum axial force (Fx) of 1^{st} storey column is -9.46×10^{-1} KN and maximum is 7.11×10^{-1} KN (figure 3.2). The minimum Displacement in x-direction (Dx) of 1^{st} storey column is -7.92×10^{-4} m and maximum is 1.05×10^{-3} m (figure 3.2). The minimum axial force (Fx) of 2^{nd} storey column is -9.46×10^{-1} KN and maximum is 7.11×10^{-1} KN (figure 3.3). The minimum Displacement in x-direction (Dx) of 2^{nd} storey column is -1.98×10^{-3} m and maximum is 2.64×10^{-3} m (figure 3.3).

Figure 3.4 and Figure 3.5 as shown below presents graphical representation of Bending Moment (My) vs Curvature (Φy) of 1^{st} storey and 2^{nd} storey column respectively. The minimum Bending Moment (My) of 1^{st} storey column is -7.11 KN-m and maximum is 9.46 KN-m (figure 3.4). The minimum Curvature (Φy) of 1^{st} storey column is -1.58×10^{-4} m^{-1} and maximum is 1.19×10^{-4} m^{-1} (figure 3.4). The minimum Bending Moment (My) of 2^{nd} storey column is -7.11 KN-m and maximum is 9.46 KN-m (figure 3.5). The minimum Curvature (Φy) of 2^{nd} storey column is -1.58×10^{-4} m^{-1} and maximum is 1.19×10^{-4} m^{-1} (figure 3.5) .

The maximum inter-storey drift for 1^{st} storey column is 1.05×10^{-4} m and maximum inter-storey drift for 2^{nd} storey column is 1.58×10^{-4} m.

Figure 3.2 Axial Force(Fx) vs Displacement in x-direction (Dx) of 1^{st} storey column.

Figure 3.3 Axial Force(Fx) vs Displacement in x-direction (Dx) of 2^{nd} storey column.

Figure 3.4 Bending Moment (My) vs Curvature (Φy) of 1^{st} storey column.

Figure 3.5 Bending Moment (My) vs Curvature (Φy) of 2^{nd} storey column.

**3.2.1.2 P-Δ Coordinate Transformation **

Similarly, by considering reinforcement ratio (ρ) 1%, P-Δ coordinate transformation is analysed and stimulated. Figure 3.6 and Figure 3.7 as shown below presents graphical representation of Axial Force (Fx) vs Displacement in x-direction (Dx) of 1st storey and 2nd storey column respectively. The minimum axial force (Fx) of 1st storey column is -7.93×10^{-1} KN and maximum is 6.8×10^{-1} KN (figure 3.6). The minimum Displacement in x-direction (Dx) of 1st storey column is -8.63×10^{-4} m and maximum is 1.01×10^{-3} m (figure 3.6). The minimum axial force (Fx) of 2nd storey column is -7.93×10^{-1} KN and maximum is 6.8×10^{-1} KN (figure 3.7). The minimum Displacement in x-direction (Dx) of 2nd storey column is -2.2×10^{-3} m and maximum is 2.6×10^{-3} m (figure 3.7).

Figure 3.8 and Figure 3.9as shown below presents graphical representation of Bending Moment (My) vs Curvature (Φy) of 1st storey and 2nd storey column respectively. The minimum Bending Moment (My) of 1st storey column is -7.57 KN-m and maximum is 8.83 KN-m (figure 3.8). The minimum Curvature (Φy) of 1st storey column is -1.5×10^{-4} m^{-1} and maximum is 1.32×10^{-4} m^{-1} (figure 3.8). The minimum Bending Moment (My) of 2nd storey column is -7.57 KN-m and maximum is 8.83 KN-m (figure 3.9). The minimum Curvature (Φy) of 2nd storey column is -1.61×10^{-4} m^{-1} and maximum is 1.38×10^{-4} m^{-1} (figure 3.9).

The maximum inter-storey drift for 1st storey column is 1.01×10^{-4} m and maximum inter-storey drift for 2nd storey column is 1.59×10^{-4} m.

Figure 3.6 Axial Force(Fx) vs Displacement in x-direction (Dx) of 1^{st} storey column.

Figure 3.7 Axial Force(Fx) vs Displacement in x-direction (Dx) of 2^{nd} storey column.

Figure 3.8 Bending Moment (My) vs Curvature (Φy) of 1^{st} storey column.

Figure 3.9 Bending Moment (My) vs Curvature (Φy) of 2^{nd} storey column.

**3.2.2 Considering Reinforcement Ratio, ρ = 2%**

Area of Steel A_{s} = 0.02*1*1

= 0.02 m^{2}

Considering 32 mm bar dia

Area of one bar =

πr2

= 3.141*(0.016)^{2} m^{2}

= 0.000804096 m^{2}

No. of Reinforcement =

AsArea of one bar

= 24

**3.2.2.1 Linear Coordinate Transformation **

By considering reinforcement ratio (ρ) 2%, linear coordinate transformation is analysed and stimulated. Figure 3.10 and Figure 3.11 as shown below presents graphical representation of Axial Force (Fx) vs Displacement in x-direction (Dx) of 1st storey and 2nd storey column respectively. The minimum axial force (Fx) of 1st storey column is -1.05 KN and maximum is 8.62×10^{-1} KN (figure 3.10). The minimum Displacement in x-direction (Dx) of 1st storey column is -4.67×10^{-4} m and maximum is 5.67×10^{-4} m (figure 3.10). The minimum axial force (Fx) of 2nd storey column is -1.05 KN and maximum is 8.62×10^{-1} KN (figure 3.11). The minimum Displacement in x-direction (Dx) of 2nd storey column is -1.17×10^{-3} m and maximum is 1.42×10^{-3} m (figure 3.11).

Figure 3.12 and Figure 3.13 as shown below presents graphical representation of Bending Moment (My) vs Curvature (Φy) of 1st storey and 2nd storey column respectively. The minimum Bending Moment (My) of 1st storey column is -8.62 KN-m and maximum is 1.05×10^{1} KN-m (figure 3.12). The minimum Curvature (Φy) of 1st storey column is -8.5×10^{-5} m^{-1} and maximum is 7.01×10^{-5} m^{-1} (figure 3.12). The minimum Bending Moment (My) of 2nd storey column is -8.62 KN-m and maximum is 1.05×10^{1} KN-m (figure 3.13). The minimum Curvature (Φy) of 2nd storey column is -8.5×10^{-5} m^{-1} and maximum is 7.01×10^{-5} m^{-1} (figure 3.13).

The maximum inter-storey drift for 1st storey column is 5.67×10^{-5} m and maximum inter-storey drift for 2nd storey column is 8.5×10^{-5} m.

Figure 3.10 Axial Force(Fx) vs Displacement in x-direction (Dx) of 1^{st} storey column.

Figure 3.11 Axial Force(Fx) vs Displacement in x-direction (Dx) of 2^{nd} storey column.

Figure 3.12 Bending Moment (My) vs Curvature (Φy) of 1^{st} storey column.

_{ }

Figure 3.13 Bending Moment (My) vs Curvature (Φy) of 2^{nd} storey column.

**3.2.2.2 P-Δ Coordinate Transformation **

Similarly, by considering reinforcement ratio (ρ) 2%, P-Δ coordinate transformation is analysed and stimulated. Figure 3.14 and Figure 3.15 as shown below presents graphical representation of Axial Force (Fx) vs Displacement in x-direction (Dx) of 1st storey and 2nd storey column respectively. The minimum axial force (Fx) of 1st storey column is -1.16 KN and maximum is 9.51×10^{-1} KN (figure 3.14). The minimum Displacement in x-direction (Dx) of 1st storey column is -5.47×10^{-4} m and maximum is 6.67×10^{-4} m (figure 3.14) . The minimum axial force (Fx) of 2nd storey column is -1.16 KN and maximum is 9.51×10^{-1} KN (figure 3.15). The minimum Displacement in x-direction (Dx) of 2nd storey column is -1.39×10^{-3} m and maximum is 1.69×10^{-3} m (figure 3.15) .

Figure 3.16 and Figure 3.17 as shown below presents graphical representation of Bending Moment (My) vs Curvature (Φy) of 1st storey and 2nd storey column respectively. The minimum Bending Moment (My) of 1st storey column is -1×10^{1} KN-m and maximum is 1.22×10^{1} KN-m (figure 3.16). The minimum Curvature (Φy) of 1st storey column is -1.01×10^{-4} m^{-1} and maximum is 8.29×10^{-5} m^{-1} (figure 3.16). The minimum Bending Moment (My) of 2nd storey column is -1×10^{1} KN-m and maximum is 1.22×10^{1} KN-m (figure 3.17). The minimum Curvature (Φy) of 2nd storey column is -1.03×10^{-4} m^{-1} and maximum is 8.46×10^{-5} m^{-1} (figure 3.17).

The maximum inter-storey drift for 1st storey column is 6.67×10^{-5} m and maximum inter-storey drift for 2nd storey column is 1.02×10^{-4} m.

Figure 3.14 Axial Force(Fx) vs Displacement in x-direction (Dx) of 1^{st} storey column.

Figure 3.15 Axial Force(Fx) vs Displacement in x-direction (Dx) of 2^{nd} storey column.

Figure 3.16 Bending Moment (My) vs Curvature (Φy) of 1^{st} storey column.

Figure 3.17 Bending Moment (My) vs Curvature (Φy) of 2^{nd} storey column.

**3.3 Case 2**

Width, b = 1.5 m

Depth, d = 1.5 m

Height, h = 10 m

Clear Cover = 5 cm

Weigth of Structure = L x B x H x Density of concrete

= 1.5 m*1.5 m*10 m*25 KN/m^{3}

= 562.5 KN

f_{c} = 40 MPa

E_{c} = 4700

√fcMPa

= 29725.41 N/mm^{2}

In UK characteristic yield stress, f_{yk} = 500 MPa

Therefore, design yield stress, f_{yd} =

fykγs

= 500/1.15 = 435 MPa

Modulus of steel may be taken to be, E_{s} = 200 GPa

**3.3.1 Considering Reinforcement Ratio, ρ = 1%**

Area of Steel A_{s} = 0.01*1.5*1.5

= 0.0225 m^{2}

Considering 24 mm bar

Area of one bar =

πr2

= 3.141*(0.016)^{2} m^{2}

= 0.000804096 m^{2}

No. of Reinforcement =

AsArea of one bar

= 28

**3.3.1.1 Linear Coordinate Transformation**

By considering reinforcement ratio (ρ) 1%, linear coordinate transformation is analysed and stimulated. Figure 3.18 and Figure 3.19 as shown below presents graphical representation of Axial Force (Fx) vs Displacement in x-direction (Dx) of 1^{st} storey and 2^{nd} storey column respectively. The minimum axial force (Fx) of 1^{st} storey column is -2.79 KN and maximum is 2.18 KN (figure 3.18). The minimum Displacement in x-direction (Dx) of 1^{st} storey column is -3.81×10^{-4} m and maximum is 4.86×10^{-1} m (figure 3.18). The minimum axial force (Fx) of 2^{nd} storey column is -2.79 KN and maximum is 2.18 KN (figure 3.19). The minimum Displacement in x-direction (Dx) of 2^{nd} storey column is -9.51×10^{-4} m and maximum is 1.21×10^{-3} m (figure 3.19).

Figure 3.20 and Figure 3.21 as shown below presents graphical representation of Bending Moment (My) vs Curvature (Φy) of 1^{st} storey and 2^{nd} storey column respectively. The minimum Bending Moment (My) of 1^{st} storey column is -2.18×10^{1} KN-m and maximum is 2.79×10^{1} KN-m (figure 3.20). The minimum Curvature (Φy) of 1^{st} storey column is -7.29×10^{-5} m^{-1} and maximum is 5.71×10^{-5} m^{-1} (figure 3.20). The minimum Bending Moment (My) of 2^{nd} storey column is -2.18×10^{1} KN-m and maximum is 2.79×10^{1} KN-m (figure 3.21). The minimum Curvature (Φy) of 2^{nd} storey column is -7.29×10^{-5} m^{-1} and maximum is 5.71×10^{-5} m^{-1} (figure 3.21) .

The maximum inter-storey drift for 1^{st} storey column is 4.86×10^{-5} m and maximum inter-storey drift for 2^{nd} storey column is 7.29×10^{-5} m.

Figure 3.18 Axial Force(Fx) vs Displacement in x-direction (Dx) of 1^{st} storey column.

Figure 3.19 Axial Force(Fx) vs Displacement in x-direction (Dx) of 2^{nd} storey column.

Figure 3.20 Bending Moment (My) vs Curvature (Φy) of 1^{st} storey column.

_{ }

Figure 3.21 Bending Moment (My) vs Curvature (Φy) of 2^{nd} storey column.

**3.3.1.2 P-Δ Coordinate Transformation**

Similarly, by considering reinforcement ratio (ρ) 1%, P-Δ coordinate transformation is analysed and stimulated. Figure 3.22 and Figure 3.23 as shown below presents graphical representation of Axial Force (Fx) vs Displacement in x-direction (Dx) of 1st storey and 2nd storey column respectively. The minimum axial force (Fx) of 1st storey column is -2.79 KN and maximum is 1.99 KN (figure 3.22). The minimum Displacement in x-direction (Dx) of 1st storey column is -3.62×10^{-4} m and maximum is 5.08×10^{-4} m (figure 3.22). The minimum axial force (Fx) of 2nd storey column is -2.79 KN and maximum is 1.99 KN (figure 3.23). The minimum Displacement in x-direction (Dx) of 2nd storey column is -9.14×10^{-4} m and maximum is 1.28×10^{-3} m (figure 3.23) .

Figure 3.24 and Figure 3.25 as shown below presents graphical representation of Bending Moment (My) vs Curvature (Φy) of 1st storey and 2nd storey column respectively. The minimum Bending Moment (My) of 1st storey column is -2.06×10^{1} KN-m and maximum is 2.89×10^{1} KN-m (figure 3.24). The minimum Curvature (Φy) of 1st storey column is -7.68×10^{-5} m^{-1} and maximum is 5.47×10^{-5} m^{-1} (figure 3.24). The minimum Bending Moment (My) of 2nd storey column is -2.06×10^{1} KN-m and maximum is 2.89×10^{1} KN-m (figure 3.25). The minimum Curvature (Φy) of 2nd storey column is -7.8×10^{-5} m^{-1} and maximum is 5.55×10^{-5} m^{-1} (figure 3.25) .

The maximum inter-storey drift for 1st storey column is 5.08×10^{-5} m and maximum inter-storey drift for 2nd storey column is 7.76×10^{-5} m.

Figure 3.22 Axial Force(Fx) vs Displacement in x-direction (Dx) of 1^{st} storey column.

Figure 3.23 Axial Force(Fx) vs Displacement in x-direction (Dx) of 2^{nd} storey column.

Figure 3.24 Bending Moment (My) vs Curvature (Φy) of 1^{st} storey column.

Figure 3.25 Bending Moment (My) vs Curvature (Φy) of 2^{nd} storey column.

**3.3.2 Considering Reinforcement Ratio, ρ = 2%**

Area of Steel A_{s} = 0.02*1.5*1.5

= 0.045 m^{2}

Considering 24 mm bar

Area of one bar =

πr2

= 3.141*(0.012)^{2} m^{2}

= 0.000804096 m^{2}

No. of Reinforcement =

AsArea of one bar

= 56

**3.3.2.1 Linear Coordinate Transformation**

By considering reinforcement ratio (ρ) 2%, linear coordinate transformation is analysed and stimulated. Figure 3.26 and Figure 3.27 as shown below presents graphical representation of Axial Force (Fx) vs Displacement in x-direction (Dx) of 1^{st} storey and 2^{nd} storey column respectively. The minimum axial force (Fx) of 1^{st} storey column is -4.37 KN and maximum is 3.77 KN (figure 3.26). The minimum Displacement in x-direction (Dx) of 1^{st} storey column is -2.59×10^{-4} m and maximum is 3.01×10^{-4} m (figure 3.26). The minimum axial force (Fx) of 2^{nd} storey column is -4.37 KN and maximum 3.77 is KN (figure 3.27). The minimum Displacement in x-direction (Dx) of 2^{nd} storey column is -6.48×10^{-4} m and maximum is 7.51×10^{-4} m (figure 3.27).

Figure 3.28 and Figure 3.29 as shown below presents graphical representation of Bending Moment (My) vs Curvature (Φy) of 1^{st} storey and 2^{nd} storey column respectively. The minimum Bending Moment (My) of 1^{st} storey column is -3.77×10^{1} KN-m and maximum is 4.37×10^{1} KN-m (figure 3.28). The minimum Curvature (Φy) of 1^{st} storey column is -4.5×10^{-5} m^{-1} and maximum is 3.89×10^{-5} m^{-1} (figure 3.28). The minimum Bending Moment (My) of 2^{nd} storey column is -3.77×10^{1} KN-m and maximum is 4.37×10^{1} KN-m (figure 3.29). The minimum Curvature (Φy) of 2^{nd} storey column is -4.51×10^{-5} m^{-1} and maximum is 3.89×10^{-5} m^{-1} (figure 3.29).

The maximum inter-storey drift for 1^{st} storey column is 3.01×10^{-5} m and maximum inter-storey drift for 2^{nd} storey column is 4.51×10^{-5} m.

Figure 3.26 Axial Force(Fx) vs Displacement in x-direction (Dx) of 1^{st} storey column.

Figure 3.27 Axial Force(Fx) vs Displacement in x-direction (Dx) of 2^{nd} storey column.

Figure 3.28 Bending Moment (My) vs Curvature (Φy) of 1^{st} storey column.

Figure 3.29 Bending Moment (My) vs Curvature (Φy) of 2^{nd} storey column.

**3.3.2.2 P-Δ Coordinate Transformation**

Similarly, by considering reinforcement ratio (ρ) 2%, P-Δ Coordinate Transformation is analysed and stimulated. Figure 3.30 and Figure 3.31 as shown below presents graphical representation of Axial Force (Fx) vs Displacement in x-direction (Dx) of 1st storey and 2nd storey column respectively. The minimum axial force (Fx) of 1st storey column is -4.6 KN and maximum is 4.09 KN (figure 3.30). The minimum Displacement in x-direction (Dx) of 1st storey column is -2.86×10^{-4} m and maximum is 3.21×10^{-4} m (figure 3.30). The minimum axial force (Fx) of 2nd storey column is -4.6 KN and maximum is 4.09 KN (figure 3.31). The minimum Displacement in x-direction (Dx) of 2nd storey column is -7.18×10^{-4} m and maximum is 8.07×10^{-4} m (figure 3.31).

Figure 3.32 and Figure 3.33 as shown below presents graphical representation of Bending Moment (My) vs Curvature (Φy) of 1st storey and 2nd storey column respectively. The minimum Bending Moment (My) of 1st storey column is -4.15×10^{1} KN-m and maximum is 4.66×10^{1} KN-m (figure 3.32). The minimum Curvature (Φy) of 1st storey column is -4.84×10^{-5} m^{-1} and maximum is 4.30×10^{-5} m-1 (figure 3.32). The minimum Bending Moment (My) of 2nd storey column is -4.15×10^{1} KN-m and maximum is 4.66×10^{1} KN-m (figure 3.33). The minimum Curvature (Φy) of 2nd storey column is -4.86×10^{-5} m^{-1} and maximum is 4.33×10^{-5} m^{-1} (figure 3.33).

The maximum inter-storey drift for 1st storey column is 3.21×10^{-5} m and maximum inter-storey drift for 2nd storey column is 4.85×10^{-5} m.

Figure 3.30 Axial Force(Fx) vs Displacement in x-direction (Dx) of 1^{st} storey column.

Figure 3.31 Axial Force(Fx) vs Displacement in x-direction (Dx) of 2^{nd} storey column.

Figure 3.32 Bending Moment (My) vs Curvature (Φy) of 1^{st} storey column.

** **

Figure 3.33 Bending Moment (My) vs Curvature (Φy) of 2^{nd} storey column.

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**CHAPTER 4.**

ANALYSIS AND DISCUSSION

**4.1 General**

This chapter deals with the analysis of the test results. It focuses on the general trends in the

results and presents the comparison of percentage increase or decrease in the values obtained.

**4.2 Analysis and discussion of test result**

- For Case 1 of 1
^{st}storey column of reinforcement ratio (ρ) 1% , maximum Axial Force (Fx) of linear coordinate transformation which is 7.11×10^{-1 }KN which is bigger than maximum Axial Force (Fx) of P-Δ coordinate transformation which is 6.8×10^{-1 }KN. Even for maximum displacement along x-direction (Dx) of linear coordinate transformation which is 1.05×10^{-3}m is bigger than maximum displacement along x-direction (Dx) of P-Δ coordinate transformation which is 1.01×10^{-3}m. Similarly for 2^{nd}storey column, maximum Axial Force (Fx) of linear coordinate transformation which is 7.11×10^{-1 }KN which is bigger than maximum Axial Force (Fx) of P-Δ coordinate transformation which is 6.8×10^{-1 }KN. Even for maximum displacement along x-direction (Dx) of linear coordinate transformation which is 2.64×10^{-3}m is bigger than maximum displacement along x-direction (Dx) of P-Δ coordinate transformation which is 2.6×10^{-3}m. - Even in Case 1 of 1
^{st}storey column of reinforcement ratio (ρ) 1% , maximum bending moment (My) of linear coordinate transformation which is 9.46 KN-m which is bigger thanmaximum bending moment (My) of P-Δ coordinate transformation which is 8.83 KN-m. However, maximum curvature (Φy) of linear coordinate transformation which is 1.19×10^{-4}m^{-1}is smaller than maximum curvature (Φy) of P-Δ coordinate transformation which is 1.32×10^{-4}m^{-1}. Similarly for 2^{nd}storey column, maximum bending moment (My) of linear coordinate transformation which is 9.46 KN-m which is bigger thanmaximum bending moment (My) of P-Δ coordinate transformation which is 8.83 KN-m. However, maximum curvature (Φy) of linear coordinate transformation which is 1.19×10^{-4}m^{-1}is smaller than maximum curvature (Φy) of P-Δ coordinate transformation which is 1.38×10^{-4}m^{-1}. - For Case 1 of 1
^{st}storey column of reinforcement ratio (ρ) 2% , maximum Axial Force (Fx) of linear coordinate transformation which is 8.62×10^{-1 }KN which is smaller than maximum Axial Force (Fx) of P-Δ coordinate transformation which is 9.51×10^{-1 }KN. Even for maximum displacement along x-direction (Dx) of linear coordinate transformation which is 5.67×10^{-4}m is smaller than maximum displacement along x-direction (Dx) of P-Δ coordinate transformation which is 6.67×10^{-4}m. Similarly for 2^{nd}storey column, maximum Axial Force (Fx) of linear coordinate transformation which is 8.62×10^{-1 }KN which is smaller than maximum Axial Force (Fx) of P-Δ coordinate transformation which is 9.51×10^{-1 }KN. Even for maximum displacement along x-direction (Dx) of linear coordinate transformation which is 1.42×10^{-3}m is smaller than maximum displacement along x-direction (Dx) of P-Δ coordinate transformation which is 1.69×10^{-3}m. - Even in Case 1 of 1
^{st}storey column of reinforcement ratio (ρ) 2% , maximum bending moment (My) of linear coordinate transformation which is 1.05×10^{1}KN-m which is smaller thanmaximum bending moment (My) of P-Δ coordinate transformation which is 1.22×10^{1}KN-m. However, maximum curvature (Φy) of linear coordinate transformation which is 7.01×10^{-5}m^{-1}is smaller than maximum curvature (Φy) of P-Δ coordinate transformation which is 8.29×10^{-5}m^{-1}. Similarly for 2^{nd}storey column, maximum bending moment (My) of linear coordinate transformation which is 1.05×10^{1}KN-m which is smaller thanmaximum bending moment (My) of P-Δ coordinate transformation which is 1.22×10^{1}KN-m. However, maximum curvature (Φy) of linear coordinate transformation which is 7.01×10^{-5}m^{-1}is smaller than maximum curvature (Φy) of P-Δ coordinate transformation which is 8.46×10^{-5}m^{-1}. - For Case 2 of 1
^{st}storey column of reinforcement ratio (ρ) 1% , maximum Axial Force (Fx) of linear coordinate transformation which is 2.18KN which is bigger than maximum Axial Force (Fx) of P-Δ coordinate transformation which is 1.99KN. Even for maximum displacement along x-direction (Dx) of linear coordinate transformation which is 4.86×10^{-4}m is smaller than maximum displacement along x-direction (Dx) of P-Δ coordinate transformation which is 5.08×10^{-4}m. Similarly for 2^{nd}storey column, maximum Axial Force (Fx) of linear coordinate transformation which is 2.18KN which is smaller than maximum Axial Force (Fx) of P-Δ coordinate transformation which is 1.99KN. Even for maximum displacement along x-direction (Dx) of linear coordinate transformation which is 1.21×10^{-3}m is smaller than maximum displacement along x-direction (Dx) of P-Δ coordinate transformation which is 1.28×10^{-3}m. - Even in Case 2 of 1
^{st}storey column of reinforcement ratio (ρ) 1% , maximum bending moment (My) of linear coordinate transformation which is 2.79×10^{1}KN-m which is smaller thanmaximum bending moment (My) of P-Δ coordinate transformation which is 2.89×10^{1}KN-m. However, maximum curvature (Φy) of linear coordinate transformation which is 5.71×10^{-5}m^{-1}is smaller than maximum curvature (Φy) of P-Δ coordinate transformation which is 5.47×10^{-5}m^{-1}. Similarly for 2^{nd}storey column, maximum bending moment (My) of linear coordinate transformation which is 2.79×10^{1}KN-m which is smaller thanmaximum bending moment (My) of P-Δ coordinate transformation which is 2.89×10^{1}KN-m. However, maximum curvature (Φy) of linear coordinate transformation which is 5.71×10^{-5}m^{-1}is smaller than maximum curvature (Φy) of P-Δ coordinate transformation which is 5.55×10^{-5}m^{-1}. - For Case 2 of 1
^{st}storey column of reinforcement ratio (ρ) 2% , maximum Axial Force (Fx) of linear coordinate transformation which is 3.77KN which is smaller than maximum Axial Force (Fx) of P-Δ coordinate transformation which is 4.09KN. Even for maximum displacement along x-direction (Dx) of linear coordinate transformation which is 3.01×10^{-4}m is smaller than maximum displacement along x-direction (Dx) of P-Δ coordinate transformation which is 3.21×10^{-4}m. Similarly for 2^{nd}storey column, maximum Axial Force (Fx) of linear coordinate transformation which is 3.77KN which is smaller than maximum Axial Force (Fx) of P-Δ coordinate transformation which is 4.09KN. Even for maximum displacement along x-direction (Dx) of linear coordinate transformation which is 7.51×10^{-4}m is smaller than maximum displacement along x-direction (Dx) of P-Δ coordinate transformation which is 8.07×10^{-4}m. - Even in Case 2 of 1
^{st}storey column of reinforcement ratio (ρ) 2% , maximum bending moment (My) of linear coordinate transformation which is 4.37×10^{1}KN-m which is smaller thanmaximum bending moment (My) of P-Δ coordinate transformation which is 4.66×10^{1}KN-m. However, maximum curvature (Φy) of linear coordinate transformation which is 3.89×10^{-5}m^{-1}is smaller than maximum curvature (Φy) of P-Δ coordinate transformation which is 4.30×10^{-5}m^{-1}. Similarly for 2^{nd}storey column, maximum bending moment (My) of linear coordinate transformation which is 4.37×10^{1}KN-m which is bigger thanmaximum bending moment (My) of P-Δ coordinate transformation which is 4.66×10^{1}KN-m. However, maximum curvature (Φy) of linear coordinate transformation which is 3.89×10^{-5}m^{-1}is smaller than maximum curvature (Φy) of P-Δ coordinate transformation which is 4.33×10^{-5}m^{-1}. - In Case 1 of reinforcement ratio (ρ) 1% , inter-storey drift of 1
^{st}storey column is almost similar for linear coordinate transformation and P-Δ coordinate transformation i.e. 1.05×10^{-4}and 1.01×10^{-4}respectively. Similarly for 2^{nd}storey column for linear coordinate transformation and P-Δ coordinate transformation, inter-storey drift is same i.e. 1.58×10^{-4}and 1.59×10^{-4}respectively. - Even in Case 1 of reinforcement ratio (ρ) 2% , inter-storey drift of 1
^{st}storey column is almost similar for linear coordinate transformation and P-Δ coordinate transformation i.e. 5.67×10^{-5}and 6.67×10^{-5}respectively. Similarly for 2^{nd}storey column for linear coordinate transformation and P-Δ coordinate transformation, inter-storey drift is same i.e. 8.50×10^{-5}and 1.02×10^{-4}respectively. - Even in Case 2 of reinforcement ratio (ρ) 1% , inter-storey drift of 1
^{st}storey column is almost similar for linear coordinate transformation and P-Δ coordinate transformation i.e. 4.86×10^{-5}and 5.08×10^{-5}respectively. Similarly for 2^{nd}storey column for linear coordinate transformation and P-Δ coordinate transformation, inter-storey drift is same i.e. 7.29×10^{-5}and 7.76×10^{-5}respectively. - In Case 2 of reinforcement ratio (ρ) 1% , inter-storey drift of 1
^{st}storey column is almost similar for linear coordinate transformation and P-Δ coordinate transformation i.e. 3.01×10^{-5}and 3.21×10^{-5}respectively. Similarly for 2^{nd}storey column for linear coordinate transformation and P-Δ coordinate transformation, inter-storey drift is same i.e. 4.51×10^{-5}and 4.85×10^{-5}respectively. - As compared to inter-storey drift of Case 1 and Case 2, inter-storey drift of Case 2 is smaller as compared to Case 1.

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**CHAPTER 5**

CONCLUSION

**5.1 General**

This present chapter deals with the conclusive results of the present study undertaken and

articulates the results in comparison with the objectives of the project study.

**5.2 Conclusion**

- For reinforcement ratio (ρ) 1%, inter-storey drift of 1
^{st}storey column for linear coordinate transformation and P-Δ coordinate transformation is almost same in both cases. Similarly, inter-storey drift of 2^{nd}storey column for linear coordinate transformation and P-Δ coordinate transformation is almost same in both case. - Even for reinforcement ratio (ρ) 2%, inter-storey drift of 1
^{st}storey column linear coordinate transformation and P-Δ coordinate transformation is almost same in both case. Similarly, inter-storey drift of 2^{nd}storey column for linear coordinate transformation and P-Δ coordinate transformation is almost same in both cases. - In Case 1 inter-storey drift decreases with increase in reinforcement ratio (ρ) for linear coordinate transformation and P-Δ coordinate transformation.
- Similarly, for Case 2 inter-storey drift decreases with increase in reinforcement ratio (ρ) for linear coordinate transformation and P-Δ coordinate transformation.
- Inter-storey drift of Case 2 is small than Case 1.

Thus it can be concluded that with increase in number of longitudinal reinforcement and dimension of the column, inter-storey drift decreases. Thus to avoid seismic action number of longitudinal bars and size of structure should be increase.

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**REFERENCE**

[1]. L. Jaillon and C.S. Poon. (2007). **Advantages and Limitations of Precast. Concrete Construction in High-rise. Buildings: Hong Kong Case Studies.** *CIB World Building Congress-011.*

[2] Eurocode 8: Seismic Design of Buildings.

[3]. C. Casotto, V.Silva et al. (2015). **Seismic**** fragility of Italian RC precast industrial structures.** *Engineering Structures Volume 94, Pages 122-136*

[4] Anze Babic and Matjaz Dolsek. (2016) **Seismic fragility functions of industrial precast building classes.** *Engineering Structures Volume 118, Pages 357-370.*

[5] Seong-Hoon Jeong and Amr S Elnashai. (2007). **Fragility relationships for torsionally-imbalanced buildings using three-dimensional damage characterization.** *Engineering Structures Volume 29, Issue 9, Pages 2172-2182.*

[6] Marianna Ercolino, Davide Bellotti et.al. (2018). **Vulnerability analysis of industrial RC precast buildings designed according to modern seismic codes.** *Engineering Structures Volume 158, Pages 67-78*

[7] Bernal D, **“Amplification factors for inelastic dynamic P-Δ effects in earthquake analysis,”** *Earthquake Engineering and Structural Dynamics, V. 15, 1987, pp. 635-51. *

[8] MacRae GA. (1994). **P−Δ effects on single-degree-of-freedom structures in earthquakes. ***Earthquake Spectra, V. 10, pp. 539-68.*

[9] Matej Fischinger, Marianna Ercolino, Miha Kramar, Crescenzo Petrone, Tatjana Isakovic. (2011). **Inelastic Seismic Shear in Multi-Storey Cantilever Columns**. *3rd ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering. *

[10] Marianna Ercolino, Crescenzo Petrone, Gennaro Magliulo, and Gaetano Manfredi. (2017). **Seismic Design of Single-StoreyPrecast Structures for P-Δ Effects. ***ACI Structural Journal. ISSN 0889-3241.*

[11] Hyo-Gyoung Kwak, Jin-Kook Kim. (2007). **P–****Δ effect of slender RC columns under seismic load**. *Engineering Structures 29, 3121–3133.*

[12] E.F. Black. (2011). **Use of stability coefficients for evaluating the P–Δ effect in regular steel moment resisting frames.** *Engineering Structures 33 1205–1216.*

[13] Eurocode 2: Design of concrete structures (EN 1992)

[14] F. Clementi , A. Scalbi, S. Lenci. (2016). **Seismic performance of precast reinforced concrete buildings with dowel pin connections.** *Journal of Building Engineering 7 224–238*

[15] Svetlana Brzev, Teresa Guevara-Perez**. Precast Concrete Construction. ***World Housing Encyclopaedia.*

[16] http://opensees.berkeley.edu

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