NONLINEAR RESPONSE OF RC FRAMES

USING PLASTIC HINGE MODEL

A THESIS SUBMITTED IN PARTIAL FULFILMENT

OF THE REQUIREMENTS FOR THE DEGREE OF

Master of Technology

in

Structural Engineering Submitted By

S SULAIMAN

Roll No. : 213CE2076

M.Tech (Structural Engineering)

Department of Civil Engineering,

National Institute of Technology

Rourkela- 769008

MAY 2015

NONLINEAR RESPONSE OF RC FRAMES

USING PLASTIC HINGE MODEL

A THESIS SUBMITTED IN PARTIAL FULFILMENT

OF THE REQUIREMENTS FOR THE DEGREE OF

Master of Technology

in

Structural Engineering

Submitted By

S SULAIMAN

Roll No. : 213CE2076

M.Tech (Structural Engineering)

Guided by

Prof. ROBIN DAVIS P

Department of Civil Engineering National Institute of Technology Rourkela

Orissa – 769008 MAY 2015

DEPARTMENT OF CIVIL ENGINEERING

NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA

ODISHA, INDIA

CERTIFICATE

This is to certify that the thesis entitled “NONLINEAR RESPONSE OF RC FRAMES USING PLASTIC HINGE MODEL”, submitted by S SULAIMAN bearing Roll no. 213CE2076 in partial fulfilment of the requirements for the award of Master of Technology in the Department of Civil Engineering, National Institute of Technology, Rourkela is an authentic work carried out by him under my supervision and guidance. To the best of my knowledge, the matter embodied in the thesis has not been submitted to any

other university/Institute for the award of any Degree or Diploma.

Place: Rourkela Prof. ROBIN DAVIS P

Date: Civil Engineering Department

National Institute of Technology, Rourkela

i

ACKNOWLEDGEMENTS

First and foremost, praise and thanks go to my God for the blessing that has bestowed upon me

in all my endeavors.

I am deeply indebted to Dr.Robin Davis, Assistant Professor, my advisor and guide, for the

motivation, guidance, tutelage and patience throughout the research work. I appreciate his broad

range of expertise and attention to detail, as well as the constant encouragement he has given me.

There is no need to mention that a big part of this thesis is the result of joint work with him,

without which the completion of the work would have been impossible.

I am grateful to Dr. Pradip Sarkar, for his valuable suggestions during the project work. I

express my most sincere admiration to my friends for their cheering up ability, optimism, and

encouragement which made this project work smoothly.

I express my sincere thanks to Prof. S. K. Sarangi, Director of National Institute of Technology,

Rourkela and Prof. S.K. Sahu, HOD, Department of Civil Engineering for their help and

providing the necessary facilities in the department.

I convey my earnest gratitude to Prof. M. R. Barik, my faculty and adviser and all faculties Prof.

A. K. Panda, Prof. A. K. Sahoo, Prof. K. C. Biswal and Prof. Asha Patel for their help in

settling down in the first year. I also thank Prof. A. V. Asha, PG Coordinator, for providing

suitable slots during the presentation and Viva.

ii

I am extremely thankful to my parents mr. S.K. Ismail and mrs. S. Begum, my brothers

S. Mubarak, S. Anwar and S. Imran for supporting me over the years and their constant moral

support and encouragement which helped me to complete of my project smoothly.

Many people have contributed to my education, and to my life, and it is with great pleasure to

accept the chance to thank them.

S Sulaiman

iii

ABSTRACT

KEYWORDS: lumped plasticity, nonlinear behavior, probabilistic analysis

Lumped plasticity hinges are popularly employed in many software packages due to easy

implementation and simplicity of the computational models. The present study is an attempt to

implement a concentrated plasticity hinge model to simulate the nonlinear behavior of RC frame

and to perform a probabilistic analysis. The model introduced by Monfortoon and Wu (1962) is

implemented in MATLAB 2012 and a probabilistic analysis is carried out by considering the

uncertainty in the modeling and geometry parameters to obtain the uncertainty in the responses.

iv

TABLE OF CONTENTS

Title Page No.

ACKNOWLEDGEMENTS………………………………………………………………….…...i

ABSTRACT ………………………………………………………………………………......…ii

TABLES OF CONTENTS…………………………………………………………………….…v

LIST OF TABLES…………………………………………………………………………..….vii

LIST OF FIGURES………………………………………………………………………….....vii

NOTATIONS…………………………………………………………………………………...ix

1 INTRODUCTION ...............................................................................................................2

1.1 Background and Motivation ..........................................................................................2

1.2 Pushover Analysis .........................................................................................................2

1.3 Objectives .....................................................................................................................3

1.4 Methodology .................................................................................................................3

1.5 Organisation of Thesis ..................................................................................................4

2 REVIEW OF LITERATURE ...............................................................................................6

2.1 General .........................................................................................................................6

2.2 Plastic Hinge Models ....................................................................................................6

2.3 Summary .................................................................................................................... 10

3 NON LINEAR STATIC ANALYSIS OF RC FRAME ...................................................... 12

3.1 General ....................................................................................................................... 12

v

3.2 Element Formulation ................................................................................................... 12

3.3 Incremental Analysis Procedure .................................................................................. 16

3.4 Steps for Modified Newton Raphson Method .............................................................. 17

3.5 Flow Chart for Modified Newton Raphson Method ..................................................... 20

3.6 Description of the Frame ............................................................................................. 21

3.7 Linear Static and Incremental Static Linear Analysis ................................................... 22

3.8 Static Nonlinear Analysis Using Modified Newton Raphson Method .......................... 23

3.9 Displacement Control Method ..................................................................................... 25

3.10 Steps for Displacement Control Method ...................................................................... 25

3.11 Flow Chart for Displacement Control Method ............................................................. 27

3.12 Static Nonlinear Analysis Using Displacement Control Method .................................. 28

3.13 Comparison of Pushover Curves with SAP2000 .......................................................... 29

3.14 Probabilistic Analysis.................................................................................................. 30

3.15 Summary .................................................................................................................... 34

4 SUMMARY AND CONCLUSIONS ................................................................................. 36

4.1 Summary .................................................................................................................... 36

4.2 Conclusions ................................................................................................................ 36

4.3 Limitations of the Study and Scope of Future Work .................................................... 37

REFERENCES ......................................................................................................................... 39

vi

Table 3.1: Se and Ce matrices for extreme values of parameter J ............................................... 15

Table 3.2: Description of the frame considered .......................................................................... 22

Table 3.3: Comparison of displacement of hinge model with exact values ................................. 22

Table 3.4: Parameters and distributions used for generation of random variables ....................... 31

Figure 3.1: Lumped plasticity model of Monfortoon and Wu (1962) ......................................... 16

Figure 3.2: Modified Newton Raphson method ......................................................................... 17

Figure 3.3: Generalized moment-rotation relationship ............................................................... 19

Figure 3.4: Flow chart for Modified Newton Raphson method .................................................. 20

Figure 3.5: Elevation of the frame and details of beam and column ........................................... 21

Figure 3.6: Lumped Plastic model of Monfortoon and Wu (1962) ............................................. 21

Figure 3.7: Static linear pushover curve ..................................................................................... 23

Figure 3.8: Moment-Rotation relationship ................................................................................. 24

Figure 3.9: Static non-linear pushover curve using Modified Newton-Raphson method ............. 24

Figure 3.10: Comparison of linear and nonlinear pushover curves ............................................. 24

Figure 3.11: Displacement control method ................................................................................ 26

Figure 3.12: Flow chart for Displacement method ..................................................................... 27

Figure 3.13: Static Non-linear pushover curve using displacement control method .................... 28

Figure 3.14: Comparison of nonlinear pushover curves ............................................................. 28

Figure 3.15: Piece wise moment rotation relationship idealized for input in SAP2000 ............... 29

Figure 3.16: Comparison of pushover curves ............................................................................. 29

Figure 3.17: Probability distributions of various input variables ................................................ 32

Figure 3.18: Histogram for maximum base shear ....................................................................... 33

vii

Figure 3.19: Pushover curves simulated using Monte-Carlo simulations .................................... 33

Figure 3.20: COV of various random variables .......................................................................... 34

viii

NOTATIONS

K elastic stiffness matrix

푆 standard elastic stiffness matrix

퐶 correction matrix

E modulus of elasticity of conctete

푟 rigidity factor

푅 connection stiffness

훥퐹 incremental load

퐷 global displacement

푑 local displacement

T transformation matrix

푟 resisting forces in local coordinates

R1 resisting forces in global coordinates

푅 residual forces in global coordinates

λ load level

Kt tangent stiffness matrix

qe external load vector

qi internal load vector

υ

1 INTRODUCTION

φ

CHAPTER-1

1 INTRODUCTION

1.1 Background and Motivation

Approaches for Non-linear analysis of RC frames can be divided into two in general, namely,

concentrated or lumped plasticity model and spread plasticity models. The concentrated

plasticity model assumes that non-linearity is lumped in at member ends. Spread plasticity

models assume that yielding starts at beam ends and that yield zone of finite length spread

inwards. Concentrated plasticity approaches are popular as it is easy to implement. Many

software packages like SAP2000 etc., implemented this approach due to the simplicity of the

computational models. Motivation of the present study is to simulate to the nonlinear behavior of

RC frame implementing a concentrated plasticity hinge and to employ it for probabilistic

analysis.

1.2 Pushover Analysis

Static push-over analysis is a simplified nonlinear analysis technique in which a structure

modelled with non-linear properties (such as plastic hinge properties) and permanent gravity

loads is subjected to an incremental lateral load from zero to a prescribed ultimate displacement

or until the structure is unable to resist further loads. The sequence of yielding, plastic hinge

formation and failure of various structural components are noted and the total force is plotted

against displacement to define a capacity curve. The analyser can monitor the behaviour of the

structure in every single load step.

χ

A pushover analysis is performed by subjecting a structure to a monotonically increasing pattern

of lateral loads that shows the inertial forces which would be experienced by the structure when

subjected to ground motion. Under incrementally increasing loads many structural elements may

yield sequentially. Therefore, at each event, the structure experiences a decrease in stiffness.

Using a nonlinear static pushover analysis, a representative non-linear force displacement

relationship can be obtained.

1.3 Objectives

Based on the literature review presented in Chapter 2, the salient objectives of the present study

have been identified as follows:

i. The objective of the study is to implement a plastic hinge model for nonlinear static

analysis of RC frames

ii. To conduct a validation study using the implemented model with the existing literatures.

iii. To do a probabilistic analysis with the implemented model.

1.4 Methodology

i. Conduct a literature review on various plastic hinge models to use in RC frame for non-

linear static pushover analysis.

ii. Identify a simple and easy to implement plastic hinge model.

iii. Implement the plastic hinge model in MATLAB 2012.

iv. Perform static non-linear analysis of RC frame and validate.

v. Consider the uncertainty of various random variables involved.

ψ

vi. Conduct a probabilistic analysis to arrive at the uncertainty in the responses such as base

shear and yield displacement. Analyse the results and arrive at conclusions.

1.5 Organisation of Thesis

Chapter 1 gives a brief introduction to the importance of the doing the nonlinear pushover

analysis. The need, objectives and scope of the proposed research work along with the

methodology is explained here.

A review on the various plastic hinge models and the methods for performing the nonlinear

analysis is discussed in the chapter2.

Development of pushover curve for the RC frame using Monfortoon and Wu (1962) hinge

model, validation of the results and the probabilistic analysis of the implemented hinge model is

explained in Chapter 3.

Finally, Chapter 4 presents discussion of results, limitations of the work and future scope of this

study.

ω

2

REVIEW OF LITERATURE

ϊ

CHAPTER 2

2 REVIEW OF LITERATURE

2.1 General

As the present study deals with non-linear static pushover analysis of the RC frames, a detailed

literature review has been conducted on modelling of RC frames and the previous work done in

the area of the lumped plasticity model has been explained in section 2.2.

2.2 Plastic Hinge Models

Monfortoon and Wu (1962) employed the rigidity-factor concept to develop a first-order elastic

analysis technique for semi-rigid frames, where the elastic stiffness matrix (K) of each member

with semi-rigid moment-connections is found as the product of the standard elastic stiffness

matrix (Se) for a member having rigid moment-connections and a correction matrix (Ce)

formulated as a function of the rigidity-factors for the two end-connections, i.e., K= Se.Ce. The

detailed formulation of Monfortoon and Wu is explained in section 3.2.

Deierlein (1991) Nonlinear behavior of steel frames was analysed by using the capacity spectrum

method. He adopted a four parametric equation to model the nonlinear moment rotation behavior

and the concentrated plasticity model incorporating both material and geometric nonlinearities.

Awkar and Lui (1999) analysed the seismic response of multistory semi-rigid frames. The

seismic responses of flexibly connected multi story frames were studied using a modified modal

analysis procedure wherein the nonlinear effects were in corporate in a fictitious force vector. It

was concluded that compared with rigidly connected frames, flexibly connected frames often

ϋ

experience larger inter story drifts in the upper stories. He noted that a more economical design

may be possible if the effect of connection flexibility is considered in the analysis.

Hasan and Grierson (2002) presented a simple computer-based push-over analysis technique for

performance-based design of building frameworks subject to earthquake loading. Used the

plasticity-factor concept that measures the degree of plastification, the standard elastic and

geometric stiffness matrices for frame elements are progressively modified to account for

nonlinear elastic–plastic behavior under constant gravity loads and incrementally increasing

lateral loads.

Hong and Wang (2003) analysed the steel frame structures and he observed that ignoring the

connection flexibility can be unconservative. He also compared the probabilities of the failure of

frames with semi-rigid connections and rigid connections. In almost all cases, the failure

probabilities of frames with semi-rigid connections appeared to be about three times less than

those of frames with semi-rigid connections.

Sekulovic et al. (2004) studied the effects of connection flexibility and material yielding on the

behavior of frames subjected to static loads are presented. To account for material yielding, a

plastic hinge concept is adopted. Based on numerical examples carried out by the developed

computer program, it can be concluded that flexible nodal connections and material yielding

based on Plastic hinge concept greatly influence frame's behavior.

Bayo et al. (2006) provided a method to model semi-rigid connections for the global analysis of

steel and composite structures. The method is based on a finite dimensioned elastic–plastic four-

node joint element.

ό

Tullini et al. (2010) vibration frequencies and mode shapes of space frames with semi-rigid

joints are analyzed using Hermitian finite elements. Several frame analysis examples were

presented to show the effects of joint flexibility and highlighted the effects of semi-rigid

connections, on the natural frequencies.

Katal (2010) rotational spring stiffness-connection ratio relation is clearly explained and revealed

in his study. A finite element program SEMIFEM is developed in FORTRAN language for the

numerical analysis. The program provides to define semi-rigid connections in terms of rotational

spring stiffness or connection ratio simultaneously. He performed numerical examples with

respect to connection percentage of the related structural members by using finite element

method. According to finite element analysis results, he found that the degree of the semi-rigid

connection is important as much as its existence in the design phase. He concluded that semi-

rigid connection should be considered to obtain more realistic, reliable and also economical

results.

Alexandre (2010) studied the free and forced nonlinear vibrations of slender frames with semi-

rigid connections are studied. An efficient nonlinear finite element program for buckling and

vibration analysis of slender elastic frames with semi-rigid connections is developed. The

equilibrium paths are obtained by continuation techniques, in combination with the Newton-

Raphson method. The results highlight the importance of the stiffness of the semi-rigid

connections on the buckling and vibration characteristics of these structures.

Tangaramvong (2011) proposed a mathematical programming based approaches for the accurate

safety assessment of semi-rigid elastoplastic frames. The inelastic behavior of the flexible

connections and material plasticity are accommodated through piecewise linearized nonlinear

ύ

yield surfaces. Solved a large number of numerical examples concerning both practical and

reasonable sized semi-rigid frames, and highlight the influence of semi-rigid connections, and

geometric nonlinearity.

Valipour (2012) formulated a novel 1D frame compound-element for the materially and

geometrically non-linear analysis of frames with flexible connections is studied. The formulation

of the element is based on the force interpolation concept and the total secant stiffness approach,

and implemented in a FORTRAN computer code. The implications of different assumptions for

the analysis and design of connections in steel frames with flush end plate and extended end

plate during different progressive collapse scenarios were studied. It was observed that the

assumption of full fixity for flush end plate and extended end plates can lead to an under

estimation of the maximum and permanent deflection as well as the axial force in the beams.

Kim et al. (2013) simple efficient numerical procedure has been developed for the nonlinear

elastic analysis of three-dimensional semi-rigid steel frames subjected to various dynamic

loadings. The geometric nonlinearity is considered by using the stability functions and geometric

stiffness matrix. He proposed a spring connection element which can be employed to model any

connection types. He found that the nonlinear semi-rigid connections dampen the deflection due

to energy dissipation and the consonance does not occur in semi-rigid frames. He also observed

that consideration of semi-rigid connections in predicting the realistic dynamic behavior of steel

structures is very much essential.

Olivares et al (2013) presented a practical numerical method, based on evolutionary computation

techniques for the optimum design of planar semi-rigid steel frames. The behavior of the semi-

rigid joints between the columns and beams is obtained by means of the Component Method. As

υτ

examples two structures are designed to demonstrate the effect of the semi-rigid connections and

drawn the following conclusions, the semi-rigid joints modify the effective column-length factor

of the members, which affects the global stability of the structure. Additionally, the semi-rigid

joints reduce the lateral stiffness of the structure.

Razavi (2014) presented the concept of the hybrid steel frame system, as it pertains to mixtures

of fully rigid and semi-rigid steel connections used in frames. Several different patterns and

locations of semi-rigid connection replacements within the frame were examined in order to

identify hybrid frames with the best seismic performance. Comparison of performance of the

rigid frames with its corresponding hybrid frames showed a superior performance for the hybrid

frame.

2.3 Summary

This Chapter discusses briefly the previous research works that uses lumped plastic model for

material nonlinear analysis of framed. A simple and convenient model for nonlinear analysis of

framed structures is required for the probabilistic analysis in this study. Though there are number

of lumped plasticity models available, the one proposed by Monfortoon and Wu (1962) is found

to be simple and easy to implement for the present study.

υυ

3 NON LINEAR STATIC ANALYSIS OF RC FRAME

υφ

CHAPTER 3

3 NON LINEAR STATIC ANALYSIS OF RC FRAME

3.1 General

This chapter deals with the development of the pushover curve of an RC portal frame using the

lumped plasticity model. A lumped plasticity model proposed by Monfortoon and Wu (1962) is

used in the present study as it is simple and a convenient technique. The nonlinear analysis of the

frame requires incremental loading and hence the incremental analysis procedures, namely,

Newton Raphson method, and Displacement Control method proposed by Batoz and Dhatt

(1979) are discussed in this Chapter. Having implemented the lumped plasticity model, a

probabilistic analysis is also carried out considering uncertainties in various parameters involved

such as Young’s modulus, moment rotation envelope and geometric parameters of the frame.

The details of the frame considered, random variables that are considered for probability analysis

and their statistical properties are also presented in this Chapter.

3.2 Element Formulation

The lumped plasticity model proposed by Monfortoon and Wu (1962) uses the rigidity-factor

concept to develop a first-order elastic analysis technique for semi-rigid frames. Elastic stiffness

matrix (K) of each member with semi-rigid moment-connections is expressed as the product of

the stiffness matrix (Se) of the elastic beam and a correction matrix (Ce) to consider the effect of

plastic hinge as shown below.

퐾 = 푆 퐶 (3.1)

The matrices (Se) and (Ce) are expressed as Eq. 3.1 and Eq.3.2 respectively.

υχ

푆 =

⎝

⎜⎜⎜⎜⎜⎛

AEL 00

−AEL 00

012EIL 6EI

L 0 −12EIL 6EI

L

0 6EIL 4AE

L 0−6EI

L 2AEL

−AEL 00

AEL 00

0 − 12EIL −6EI

L 0 12EIL −6EI

L

0 6EIL 4AE

L 0−6EI

L 2AEL ⎠

⎟⎟⎟⎟⎟⎞

(3.2)

퐶 =( )

⎣⎢⎢⎢⎢⎡

e 000000e e 0000e e 000000e 00

0000e e0000e e ⎦

⎥⎥⎥⎥⎤

(3.3)

The elements of the matrix, Ce are defined in terms of rotational stiffness of the spring as given

below.

푒 = 4 − 푟 푟 (3.3.a) 푒 = 3푟 (2− 푟 ) (3.3.f)

푒 = 4푟 − 2푟 + 푟 (3.3.b) 푒 = −2퐿푟 (1− 푟 ) (3.3.g)

푒 = 3푟 (2 − 푟 ) (3.3.c) 푒 = 6/퐿(푟 − 푟 ) (3.3.h)

푒 = 4− 푟 푟 (3.3.d) 푒 = 6/퐿(푟 − 푟 ) (3.3.i)

푒 = 4푟 − 2푟 + 푟 푟 (3.3.e) 푒 = 2퐿푟 (1− 푟 ) (3.3.j)

Where 푟 푎푛푑푟 are the rigidity factors at joints 1 and joint 2 (see Fig. 3.1). The rigidity-factor

푟 defines the rotational stiffness of the joint relative to that of the attached member and (as

shown in Fig 3.1) can be defined as the ratio of the end-rotation (훼 ) of the member to the

combined rotation (휃 ) of the member and the joint.

υψ

Monfortoon and Wu (1962) express the rigidity factor (ri) in terms of modulus of elasticity (E),

moment of inertia (I) and length of the member (L) and rotational stiffness of the spring (J). It

can be observed from Eq. (3.4) that value of ri will be zero, when Ji is infinity and ri will be unity

when Ji is zero.

푟 = = (푖 = 1,2) (3.4)

In general ri may take values between zero and one for the remaining values of J. The joint

becomes perfectly rigid when rotational stiffness is infinity and it becomes pinned when

rotational stiffness is zero.

Table 3.1 presents the stiffness matrices Ce, Se and K for Ji = 0 and Ji = infinity. It can be seen

that as Ji becomes infinity, the element stiffness will be same as that of elastic beam. As the

nonlinear analysis using this lumped plasticity element requires incremental analysis the standard

procedures followed for the same are explained in the next sections.

υω

Table 3.1: Se and Ce matrices for extreme values of parameter J

J Se Ce

Infinity

퐴퐸퐿

0 0 −퐴퐸퐿

0 0 1 0 0 0 0 0

0 12퐸퐼퐿

6퐸퐼퐿

0 −12퐸퐼퐿

6퐸퐼퐿

0 1 0 0 0 0

0 6퐸퐼퐿

4퐸퐼퐿

0 −6퐸퐼퐿

2퐸퐼퐿

0 0 1 0 0 0

−퐴퐸퐿

0 0 퐴퐸퐿

0 0 0 0 0 1 0 0

0 −12퐸퐼퐿

−6퐸퐼퐿

0 12퐸퐼퐿

−6퐸퐼퐿

0 0 0 0 1 0

0 6퐸퐼퐿

4퐸퐼퐿

0 −6퐸퐼퐿

2퐴퐸퐿

0 0 0 0 0 1

Zero

퐴퐸퐿

0 0 −퐴퐸퐿

0 0 1 0 0 0 0 0

0 12퐸퐼퐿

6퐸퐼퐿

0 −12퐸퐼퐿

6퐸퐼퐿

0 0 0 0 0 0

0 6퐸퐼퐿

4퐸퐼퐿

0 −6퐸퐼퐿

2퐸퐼퐿

0 0 0 1 0 0

−퐴퐸퐿

0 0 퐴퐸퐿

0 0 0 0 0 0 0 0

0 −12퐸퐼퐿

−6퐸퐼퐿

0 12퐸퐼퐿

−6퐸퐼퐿

0 0 0 0 0 0

0 6퐸퐼퐿

4퐸퐼퐿

0 −6퐸퐼퐿

2퐴퐸퐿

0 0 0 0 0 1

υϊ

Figure 3.1: Lumped plasticity model of Monfortoon and Wu (1962)

3.3 Incremental Analysis Procedure

Modified Newton-Raphson method (MNR) is used in the present study for incremental analysis.

It evaluates the out-of-balance load vector, which is the difference between the internal forces

(the loads corresponding to the element stresses) and the applied loads. The program then

performs a linear solution as explained in section 3.4 and checks for convergence. If convergence

criteria is not satisfied, the out-of-balance load vector is re-evaluated. This iterative procedure

continues until the problem converges. The solution algorithm for the Modified Newton Raphson

method is explained graphically in Fig. 3.2.

(a) Lumped plasticity model (b) Degree of freedom

(c) Element idealization to elastic beam and spring

υϋ

3.4 Steps for Modified Newton Raphson Method

The step wise iterative procedure for the Modified Newton Raphson method for the lumped

plasticity model as explained below is implemented in a MATLAB 2012 environment. A flow

chart is provided in the Fig. 3.4.

1. Apply the initial force 훥푭ퟎ (refer the Fig. 3.2).

2. Find the global displacements from the algebraic equation .

푫ퟏ = 퐾 훥푭ퟎ (3.5)

Where D1 is the displacement vector in global coordinates and K is the stiffness matrix.

3. Calculate the local displacement of each element by applying transformations.

풅ퟏ = 푇푫ퟏ (3.6)

Where d1 is the displacement vector in local coordinates.

T is the transformation matrix.

Figure 3.2: Modified Newton Raphson method

퐷3

퐷

퐷2 훥퐷 훥퐷2 Displacement D

Forc

e ΔF

0

υό

4 At each node, the beam and the spring undergo same rotation (Eq 3.7), and the moment

resisted at each node is equal to the sum of the moment resisted by the beam and the

spring (Eq 3.8).

푑 = 푑 (3.7)

푟 (푑 ) = 푟 (푑 ) + 푟 (푑 ) (3.8)

Where dθb and dθs are the rotations in the beam and spring in local coordinates

respectively.

r1(d1) is the internal force at displacement d1 at a node.

rb(d1) and rs(d1) are the internal forces in beam and spring at displacement d1

respectively.

5 The nonlinearity in the spring is introduced from the moment-rotation relationship of the

joint. A generalised moment rotation relationship is shown in Fig 3.3.

푟 (푑 ) = 푟 (푑 ) + 푟 (푑 ) (3.9)

Where 푟 (푑 ) is the internal force at displacement d1 after introducing the nonlinearity in

the spring at a node.

푟 (푑 ) is the nonlinearity in the spring.

6 Transform the element forces to global forces.

푹ퟏ(퐷 ) = 푇 풓ퟏ (3.10)

Where 푹ퟏ(퐷 ) Is the internal force vector in global coordinates.

7 Calculate the residual forces and check for the convergence.

푹ퟏ(퐷 ) = 푹ퟏ(퐷 ) − 훥푭ퟎ (3.11)

Where 푹ퟏ(퐷 ) is the residual force vector.

υύ

8 If the convergence is not achieved, calculate the incremental displacement for the next

iteration.

∆푫ퟏ = 퐾 푹ퟏ (3.12)

Where ∆푫ퟏ is the incremental displacement vector.

9 Update the displacement field.

푫ퟐ = 푫ퟏ + ∆푫ퟏ (3.13)

10 The steps are repeated until convergence is achieved.

Figure 3.3: Generalized moment-rotation relationship

Mom

ent (푟

)

φτ

3.5 Flow Chart for Modified Newton Raphson Method

Apply the initial force 퐹

푈1 = 퐾 퐹 Calculate the global displacements using

Transform global displacements to local displacement

푟 (푑 ) = 푟 (푑 ) + 푟 (푑 ) Introduce the nonlinearity from moment-rotation relationship Fig 3.3

The total force at each node is the sum of the force resisted by beam and spring i.e., 푟 (푑 ) = 푟 (푑 ) + 푟 (푑 )

Transform the element force to global forces (R) and calculate residual force 푅 = 푅–퐹

Check for convergence Norm(r)<error

Update displacement using 푈2 = 푈1 + 훥푈

Calculate 훥푈 = 퐾 푅

Create vectors containing displacement and load

Print the load-displacement configuration

Stop

Yes

No

Figure 3.4: Flow chart for Modified Newton Raphson method

φυ

3.6 Description of the Frame

In this study an RC portal frame with 3.2m height and 5m span. The elevation of the frames is

shown in Fig. 3.5 a. The section details of the beam and column are shown in Fig 3.5 b and Fig

3.5 c. The base of the frame is considered as fixed. The material and geometric properties

considered for the frame is given in Table 3.2. The modeled frame as per Monfortoon and Wu

(1962) is shown in the Fig 3.6. The frame is analyzed for incremental loading as per

implemented lumped plasticity model.

(a) Elevation

Figure 3.5: Elevation of the frame and details of beam and column

(b) Beam (c) Column

Figure 3.6: Lumped Plastic model of Monfortoon and Wu (1962)

X

Y

φφ

Table 3.2: Description of the frame considered

3.7 Linear Static and Incremental Static Linear Analysis

In order to check the convergence of linear analysis problem, a linear static analysis of the frame

is conducted by assigning Ji values for all the joints as infinity in the implemented lumped

plasticity model. Linear static analysis of the frame is carried for a lateral load 10kN (at node 2

along the X axis) and the displacement obtained is compared with exact values. The

displacements and rotations at node 2 and node 3 are matching with exact values with negligible

errors. A linear static pushover analysis is also for the same frame and a pushover curve is

obtained as shown in Fig 3.7.

Table 3.3: Comparison of displacement of hinge model with exact values

Node Present study Exact results

Node Displacement (mm) Rotation (rad) Displacement (mm) Rotation (rad)

2 1.436 -3.821x10-4 1.4362 -3.8213x10-4

3 1.436 -3.821x10-4 1.4362 -3.8213x10-4

Element No Size(m) Area(m2) Moment of inertia (m4) Young’s modulus (kN/m2)

1 0.3x0.3 0.09 6.75x10-4 25x106

2 0.25x0.3 0.075 5.625x10-4 25x106

3 0.3x0.3 0.09 6.75x10-4 25x106

φχ

3.8 Static Nonlinear Analysis Using Modified Newton Raphson Method

Static nonlinear analysis has been done for the frame (details shown in Table 3.2) in MATLAB

2012 using modified Newton Raphson method as explained section 3.4 and 3.5. The

Moment(M)-Rotation(θ) relation Eq.3.14.

푀 = 퐾 휃 − 퐾 휃 + 퐾 휃 (3.14)

is assumed as cubic curve shown in Fig 3.8. Pushover curve is developed using MATLAB 2012

and the curve obtained is shown in Fig 3.9. Comparison of the pushover curve for static linear

and static non-linear case is done and shown in Fig 3.10.It can be seen from the Fig 3.9 that

Modified Newton Raphson method fails at the peak points and hence it is difficult to get the

entire load-displacement response after the peak. In order to overcome this problem,

displacement control method is used. The algorithm for Displacement Control Method proposed

by Batoz and Dhatt (1979) is explained in section 3.7.

0

10

20

30

40

50

60

0 0.002 0.004 0.006 0.008

Forc

e kN

Displacement m

Figure 3.7: Static linear pushover curve

퐾 = 1푋10

퐾 = 12푋10

퐾 = 38푋10

where

φψ

0

10

20

30

40

0 0.01 0.02 0.03 0.04 0.05M

omen

t kN

-m

Rotation rad

05

10152025303540

0 0.001 0.002 0.003 0.004

Forc

e kN

Displacement m

01020304050607080

0 0.002 0.004 0.006 0.008 0.01 0.012

Forc

e kN

Displacement m

Figure 3.8: Moment-Rotation relationship

Figure 3.9: Static non-linear pushover curve using Modified Newton-Raphson method

Figure 3.10: Comparison of linear and nonlinear pushover curves

φω

3.9 Displacement Control Method

In this section the Standard Incremental Displacement Algorithm Fig.3.11 proposed by Batoz

and Dhatt (1979) has been discussed. The numerical problem that may be encountered when

tracing the nonlinear load-deflection curve is the ill-conditioning of the tangent stiffness matrix

near the peak point. This may lead to numerical over-flow and divergence in the computer

analysis. In case of divergence, it is difficult to figure out whether the system is failed

structurally or numerically. To overcome this difficulty displacement method is used, it gives a

good solution for nonlinear problems because it presents a great stability at the critical points.

Also, it presents the adaptability of the load parameter, which reflects variations in the stiffness,

and the ability to determine the direction of load automatically.

3.10 Steps for Displacement Control Method

1. Let (d0, λ0) be the equilibrium initially at i=0 (d0is the initial displacement vector and λ0 is

the load level)

2. qth component of d0 is incremented by δd(q). Alter the initial displacement vector d0 such

that 푑 (푞) = 푑 (푞) + 훿푑(푞)

3. Calculate the residual vector 풓 = 풒 − 흀풒

Where 풒 is the internal load vector.

흀 is the load level parameter.

풒 is the external load vector.

4. Find the displacement vectors 훿풅 and 훿풅

훿풅 = 퐾 풓 and 훿풅 = 퐾 풒

φϊ

5. Calculate the incremental load level and incremental displacement 훿휆 푎푛푑훿풅

훿풅 = 훿풅 + 훿휆훿풅 and 훿휆 = ( )( )

6. Displacement vector and the load level are updated

퐝 = 퐝 + δ풅 and 휆 = 휆 + 훿휆

7. Repeat the steps until a desired accuracy or desired number of iterations are achieved.

훿푑(푞) 훿푑(푞) 훿푑(푞)

λ1 λ2 λ3

Displacement

Load level

Figure 3.11: Displacement control method

φϋ

3.11 Flow Chart for Displacement Control Method

Standard Incremental Displacement Algorithm proposed by Batoz and Dhatt (1979) is shown in

the flow chart given below.

Give initial force 풒

Find displacement vector d0

qth component of d is incremented by δd(q).

Calculate the residue vector 풓

Compute displacement vectors 훿풅 and 훿풅 .

Compute 훿휆 푎푛푑훿풅

퐝 = 퐝 + δ풅 and 휆 = 휆 + 훿휆

Create vectors containing displacement and load

Print the load-displacement configuration

Stop

Check for convergence Norm(r)<error

Figure 3.12: Flow chart for Displacement method

φό

3.12 Static Nonlinear Analysis using Displacement Control Method

Pushover analysis is carried using displacement control method proposed by Batoz and Dhatt

(1979) and the corresponding pushover curve obtained is shown in Fig 3.13. Comparison of

pushover curves obtained by Modified Newton Raphson method and the displacement control

method is shown Fig.3.14. It can be seen that Newton Raphson method failed after passing the

peak point where as the displacement control method could predict the post peak behavior also.

0

5

10

15

20

25

30

35

40

0 0.02 0.04 0.06 0.08 0.1

Forc

e kN

Displacement m

0

5

10

15

20

25

30

35

40

0 0.01 0.02 0.03

Forc

e kN

Displacement m

Modified NewtonRaphson Method

DisplacementControl Method

Figure 3.13: Static Non-linear pushover curve using displacement control method

Figure 3.14: Comparison of nonlinear pushover curves

φύ

3.13 Comparison of Pushover Curves with SAP2000

A validation study is carried out to check the accuracy of the implemented model. Static

nonlinear analysis has been done for the frame (details shown in Table 3.1) in SAP2000 and

input the moment-rotation relationship used in SAP2000 is shown in Fig 3.15. The pushover

curves obtained from SAP2000 and the present study are shown in Fig 3.16. It can be seen that

pushover curves are fairly matching with negligible error in the yield level. This may be due to

the piecewise idealization of the input moment rotation curves.

0

5

10

15

20

25

30

35

40

0 0.02 0.04 0.06

Mom

ent k

N-m

Rotation rad

Idealisedmomentrotation curve

Parabolicmomentrotation curve

05

10152025303540

0 0.01 0.02 0.03

Forc

e kN

Displacement m

SAP2000

Present Study

Figure 3.15: Piece wise moment rotation relationship idealized for input in SAP2000

Figure 3.16: Comparison of pushover curves

χτ

3.14 Probabilistic Analysis

Uncertainties associated with geometric tolerances, material properties, boundary conditions and

etc., widely exist in practical engineering problems. A probability analysis is performed in

MATLAB 2012 for the implemented plastic hinge model. In this study, only variables that are

considered to have a significant effect on the pushover curve are selected as random variables.

Selected variables are Young’s modulus (E) for material variability of concrete, cross sectional

dimensions of the beam and column, and the constants of the moment-rotation relationship

(퐾 ,퐾 푎푛푑퐾 ). Statistical properties of each random variable are given in Table 3.4. Probability

analysis is done by taking 10000 samples. A plot of the probability distribution of various input

variables and the histogram for base shear obtained from probability analysis are shown in the

Fig.3.17 and Fig.3.18. Pushover curves generated by Monte-Carlo simulation incorporating the

uncertainty in the input variables are displayed in the Fig.3.19. The uncertainty in the capacities

of base shear and top displacements can be clearly observed from this plot. It can be seen from

Fig.3.20 that a C.O.V of 8.64% is observed in the maximum base shear due to the uncertainty in

young’s modulus of concrete (7.60%), moment-rotation constants (2.75%), the width of the

section (0.11%) and depth of the section (0.90%).

χυ

Table 3.4: Parameters and distributions used for generation of random variables

Random variable Parameters Probability distribution

function Mean C.O.V.

(%) Reference

Young’s modulus of concrete E Lognormal 25x106 kN/m2 7.6 Devandiran

et.al.(2013)

Moment-rotation constants

K1 K2 K3

Lognormal 1x106 kN-m

12x104 kN-m 38x102 kN-m

2.75 2.75 2.75

Hyung Lee et al.(2006)

Beam Width Depth Lognormal 0.25 m

0.3 m 0.108

0.9 Dimitri

et.al(1997)

Column Width Depth Lognormal 0.3 m

0.3 m 0.108

0.9 Dimitri

et.al(1997)

χφ

0.24 0.245 0.25 0.255 0.26 0.2650

50

100

150

200

250

300

350

(a) Width of beam

Freq

uenc

y

0.29 0.295 0.3 0.305 0.31 0.315 0.320

50

100

150

200

250

300

350

(b) Depth of beam

Freq

uenc

y0.24 0.245 0.25 0.255 0.26 0.2650

50

100

150

200

250

300

350

(c) Width of column

Freq

uenc

y

0.285 0.29 0.295 0.3 0.305 0.31 0.3150

100

200

300

400

(d) Depth of column

Freq

uenc

y

Figure 3.17: Probability distributions of various input variables

0 0.5 1 1.5 2 2.5x 10

6

0

100

200

300

400

(e) Moment rotation constant K1

Freq

uenc

y

0.5 1 1.5 2 2.5x 10

5

0

100

200

300

400

(f) Moment rotation constant K2

Freq

uenc

y

0 2000 4000 6000 80000

100

200

300

400

(g) Moment rotation constant K3

Freq

uenc

y

1.5 2 2.5 3 3.5x 10

7

0

100

200

300

400

(h) Youngs modulus of concrete E

Freq

uenc

y

χχ

25 30 35 40 45 50 550

50

100

150

200

250

300

350

Base Shear kN

Freq

uenc

y

Mean=37.11 kN

Standard deviation=3.21 kN

C.O.V=8.64%

Figure 3.18: Histogram for maximum base shear

Figure 3.19: Pushover curves simulated using Monte-Carlo simulations

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

10

20

30

40

50

60

Displacement m

Bas

e sh

ear k

N

χψ

3.15 Summary

The first part of this chapter presents modeling of the frame for nonlinear analysis. The next part

of the chapter explains the numerical methods, Newton Raphson method and displacement

method which have been used for nonlinear analysis. Newton Raphson method becomes unstable

after passing through limit points and fails to follow the equilibrium path. To overcome this

problem displacement control method has been used. The later part of the chapter presents

pushover curves (static linear and static non-linear) for the modeled RC frame. In the last part of

the chapter validation of the study and the probabilistic analysis is presented.

0 5 10

Young's Modulus

Moment-Rotation constants

Width

Depth

Base Shear

COV in percentage

Figure 3.20: COV of various random variables

χω

4 SUMMARY AND CONCLUSIONS

χϊ

CHAPTER 4

4 SUMMARY AND CONCLUSIONS

4.1 Summary

The objective of the study is to implement a plastic hinge model for nonlinear static analysis of

RC frames for conducting a probabilistic study. The plastic hinge model of Monfortoon and Wu

(1962) has been implemented in MATLAB 2012. Nonlinear static analysis of implemented

plastic hinge model has been carried out using load control Modified Newton Raphson method

and displacement control method proposed by Batoz and Dhatt (1979). The pushover curve

obtained from the present study is validated using that of SAP2000. A probabilistic analysis is

carried out using the implemented lumped plasticity model considering uncertainties in the

parameters such as Young’s modulus (E) for material variability of concrete, cross sectional

dimensions of the beam and column, and the moment-rotation constants. The conclusions

obtained from the study, limitations of the present work and the future scopes of this research are

given below.

4.2 Conclusions

The following are the major conclusions from the present study:

Load control Newton Raphson method failed to yield the entire load –displacement

response in the post-peak region.

The Displacement Control method could yield the post peak behaviour of the pushover

curve.

χϋ

The base shear versus displacement curves obtained from the lumped plasticity model

and SAP2000 are fairly matching. Hence the implemented model in MATLAB 2012 is

used for Probabilistic analysis.

Probability analysis shows that there is uncertainty in the values of maximum base shear

indicated by the COV value of about 8.64%. This uncertainty is due to the uncertainty in

the input parameters such as Young’s modulus of concrete (7.6%), moment-rotation

constants (2.75%), the width of the section (0.108%), and depth of the section (0.9%).

4.3 Limitations of the Study and Scope of Future Work

The limitations of the present study are summarised below.

The present study considered only the lumped plasticity model.

Only material nonlinearity is considered the geometric nonlinearity is not considered.

Uncertainties in modelling are considered only for Young’s modulus of concrete,

geometric properties of the section and moment-rotation constants, the other uncertainties

like density of concrete, length of the member can be incorporated in future work.

The present study incorporates only lumped plasticity model it can be extended for

distributed plasticity model in future work.

χό

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χύ

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