5-1 FE Analysis of RC Frames

Chapter 3 Finite Element Analysis of RC Frames

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Chapter 3

Finite Element Analysis of RC Frames

3.1 Introduction

The finite element method (FEM) of analysis has developed into the most widely

used method and tool for engineers and researchers. It provides the engineer with the

capability to analyze complex structural system in a much more realistic manner

regarding the geometry (Noguchi and Schnobrich 1993) and the loading and support

conditions. For nonlinear analysis of RC frames, many finite element (FE) models have

been developed since Ngo and Scordelis (1967) published the earliest application of FEM

to RC beams. According to Miramontes et al. (1996), these FE models can be generally

divided into three categories: local models (microscopic finite element models), global

models (member models), and semi-local models (fiber models), as introduced in Chapter

1.

Local models use a continuous media approach (Miramontes et al. 1996). Structural

members as well as joints are discretized into a large number of finite elements.

Mathematical models describing the stress-strain relationships of concrete and steel,


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bond-slip effect between steel bar and the surrounding concrete, shear-sliding effect

between steel bar and the cracked surface, opening and closing of cracks, etc., are

expressed in local variables. Since models of this type require the solution of a large

system of equations, they are not suitable for dynamic analysis or cyclic static analysis at

the structural level. Local models are typically applied to analysis of such local behavior

as member or joint response and are not in the scope of this research.

The following sections of this chapter deal with the literature survey of global

models and semi-local models and the description of the state-of-the-art FE models for

dynamic and cyclic static analysis of RC frame structures. The background of previous

global and semi-local models for RC frame element is introduced and several

representative models are described in more details. The derivation of the force-based

fiber element model employed in this research is also presented.

3.2 Global Models

The global model reported herein is defined as the model in which an RC member

(beam or column) is modeled with a single two-node line element or several two-node

line elements, and the element stiffness matrix is obtained by summing the weighted

contributions of the member-wise sectional characteristics which are not stored and thus

cannot be traced back. The Navier-Bernoulli hypothesis (the hypothesis that plane

sections remain plane) is usually assumed for the calculation of the sectional


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characteristics of the member. Also, only the envelopes of the material stress-strain

curves are usually taken into account, and hence an ad-hoc set of phenomenological rules

for unloading and reloading response of section or member usually has to be incorporated.

The general solution procedure of dynamic analysis using global models is illustrated in

Fig.3.1.

Some of the phenomenological hysteretic models found from the literature are

shown in Fig.3.2, and the comparison of certain features among these models is indicated

in Table 3.1. Global models can be further classified into lumped models and distributed

models.

Fig. 3.1 Illustration of the solution procedure for global models (Mo 1994).


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Fig. 3.2 Models for hysteresis loops (Mo 1994).


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Table 3.1 Comparison of hysteretic models (Mo 1994)


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3.2.1 Lumped Nonlinearity Models

As the inelastic behavior is often concentrated at the ends of beams and columns in

frames under seismic excitation, an early approach to modeling the nonlinear behavior of

RC members was by means of nonlinear rotational springs located at the member ends

(Zeris 1986). These models are referred to as lumped nonlinearity models. The earliest

lumped model was formally proposed by Gibson (1967), although it had been reportedly

used earlier. It consisted of an elastic element with two nonlinear rotational springs

connected at both ends of the elastic element and was thus also referred to as one

component model (Otani 1974). The configuration of this model is schematically

illustrated in Fig.3.3. The inelastic deformations of the member are lumped into the end

springs. The inelastic moment-rotation relationship of a spring was determined assuming

the point of inflection at the center of the member. For instance, Suko and Adams (1971)

determined the spring stiffness based on the location of the inflection point at the initial

elastic stage. This model is versatile in that various sources of nonlinearity can be

specified by addition of corresponding nonlinear springs. Thus, the phenomenological

constitutive relationship for lumped models can be easily incorporated in such models.

Nonlinear rotational springs

Fixed inflection pointElastic member EI

Fig. 3.3 Lumped nonlinearity model (Giberson 1967).


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Review of several lumped spring constitutive models has been reported in Zeris

(1986) and Taucer (1991), and is summarized as follows: Such lumped plasticity

constitutive models include cyclic stiffness degradation in flexure and shear (Clough and

Benuska 1966, Takeda etal. 1970, Brancaleoni et al. 1983), pinching under reversal

(Banon et al. 1981, Brancaleoni et al. 1983, D’Ambrisi and Filippou 1999) and fixed end

rotations at the beam-column joint interface due to bar pull-out (Otani 1974, Filippou and

Issa 1988, D’Ambrisi and Filippou 1999). Nonlinear rate constitutive representations

have also been generalized from the basic endochronic theory (Bazant and Bhat 1977,

Rivlin 1981, Valanis 1981) formulation in Ozdemir (1981) to provide continuous

hysteretic relations for the nonlinear springs. An extensive discussion on the

mathematical functions that are appropriate for such models was given in Iwan (1978).

An interesting and perhaps one of the most sophisticated lumped models was

proposed by Lai et al. (1983). This model consisted of two inelastic zero-length

subelement at the ends of a RC member sandwiching a linear elastic line element, as

shown in Fig.3.4. For each inelastic subelement, there are four inelastic corner springs

and one center spring. Each of the corner springs represents the stiffness of the effective

reinforcing steel bars and the effective compression concrete. The center spring

represents the effective concrete in the center region and is only effective when in

compression. This model can be regarded as a fiber hinge model and was found to be able

to simulate the axial force-biaxial bending interaction in a more rational way than is

possible by classical plasticity theory as mentioned in section 3.2.3.

The basic advantage of the lumped model is certainly its simplicity that reduces

computational requirement and improves numerical stability. Most lumped models,

however, overlook certain aspects of the hysteretic behavior of RC members and are,


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therefore, limited in applicability. For instance, the parametric and theoretical studies

presented by Anagnostopoulos (1981) demonstrated a strong dependency of the model

parameters on the imposed loading pattern and the level of inelastic deformation, which

revealed the sources of the hindrance to modeling the nonlinear behavior with two

zero-length end springs or elements. Also, lumped models are generally unable to account

for the deformation softening behavior of RC members. There are some other issues with

lumped element models as well as the lumped spring constitutive models. Some of the

issues are actually common to distributed nonlinearity models as well. These common

issues are to be discussed in the later sections with distributed models.

3.2.2 Distributed Nonlinearity Models

The realistic nonlinear behavior can spread over a finite length of the member rather

than concentrates at a point. The other approach to modeling the hysteretic behavior of

RC member is, thus, to capture the global behavior of the member by weighted

integration of the responses of several monitoring cross sections along the member or by

Fig. 3.4 Fiber hinge lumped model (Lai et al. 1983).


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means of certain inelastic sub-elements of finite length.

One of the earliest distributed models was introduced by Clough and Johnston

(1966). The model consists of two parallel elements, one elastic-perfectly plastic to

represent yielding and the other perfectly elastic to represent strain-hardenig. The

ensemble element allows for a bilinear moment-rotation relation along the member. It

was referred to as two-component element (Otani 1974).

Otani (1974) presented an element model that consisted of two parallel flexible line

sub-elements (linearly elastic and inelastic) and two inelastic rotational springs at the

ends of the flexible line sub-elements. The inelastic deformations were lumped in the

rotational springs as in the lumped models, while the global behavior of the member is

derived by integration of the curvatures along the two parallel sub-elements. A fixed point

of contraflexure was assumed in the derivation of the stiffness matrix of this model,

which is supposed to be the main limitation of this and similar models.

Soleimani et al. (1979) proposed a model in which an inelastic zone spreading from

the beam-column interface into the member as a function of loading history was first

introduced. A similar model was developed by Meyer et al. (1983) and was later extended

by Roufaiel and Meyer (1987) to include the shear effect and the axial force effect based

on a set of empirical rules. Roufaiel and Meyers’ model is described in details in the

following paragraph.

An element is subdivided into three segments as shown in Fig.3.5: 1) an inelastic

segment of length Xi at node i, with the average stiffness (EI) i; 2) an inelastic segment of

length Xj at node j, with the average stiffness (EI) j.; and 3) a centered elastic segment of

length (L - Xi - Xj), with the initial elastic stiffness (EI) e. The (EI) e and (EI) i values are

obtained from the simplified bilinear moment-curvature envelope curve of the


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corresponding cross sections together with a modified Takeda hysteretic

moment-curvature relationships (Takeda et al. 1970, Popov et al. 1972, Ma et al. 1976) as

shown in Fig.3.5. The stiffness ratio is defined as Qi = (EI) e / (EI) i. The length Xi, Xj and

stiffness ratio Qi of the inelastic region at node i depend on the current branch of

moment-curvature diagram. The element stiffness matrix, then, can be calculated from

equations (3.1):

66656362

56555352

4441

36353332

26252322

1411

0000

00000000

0000

kkkkkkkk

kkkkkkkkkk

kk

kij , (3.1a)

where

LEA

kkkk 41144411 (3.1b)

2122211

2222 AAA

Ak

(3.1c)

2122211

1223 AAA

Ak

(3.1d)

2332 kk (3.1e)

2122211

1133 AAA

Ak

(3.1f)

2252 kk (3.1g)

2353 kk (3.1h)

322262 kLkk (3.1i)

332363 kLkk (3.1j)


3-11

2122211

2255 CCC

Ck

(3.1k)

2122211

1256 CCC

Ck

(3.1l)

5665 kk (3.1m)

5525 kk (3.1n)

5626 kk (3.1o)

655535 kLkk (3.1p)

665636 kLkk (3.1q)

33311 )1()()1(

)(31

LQQXLQXEI

A jjjiie

(3.1r)

22212 1)(1

)(21

LQQXLQXEI

A jjjiie

(3.1s)

LQQXLQXEI

A jjjiie

1)(1)(

122 (3.1t)

33311 1)(1

)(31

LQQXLQXEI

C iiijje

(3.1u)

22212 1)(1

)(21

LQQXLQXEI

C iiijje

(3.1v)

LQQXLQXEI

C iiijje

1)(1)(

122 (3.1x)


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Node jNode i

(EI)(EI)

MM

M M

Xi Xj

i

i

j

je

y

y

L

1

2

34

5

6(EI)

Fig. 3.5 Modeling with finite-length inelastic zone (Meyer et al. 1983).

M

1

2 8

397

13

6

11 5

10

124

M

1

2

3

4

2

3

3

4

Fig. 3.6 Hysteretic moment-curvature relationship (Takeda et al. 1970).


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Darvall and Mendis (1985) proposed a similar but simpler model with end inelastic

deformations defined through a trilinear moment-curvature relation (Taucer et al. 1991).

Filippou and Issa (1988) and D’Ambrisi and Filippou (1999) also subdivided the element

in several subelements, but followed a different approach. Each effect, such as spread

plasticity, interface bond-slip, shear, was modeled by a subelement. All the subelements

were connected in series and/or in parallel to simulate the overall behavior of the member

in a similar manner as that of Otani’s model (1974).

3.2.3 Limitations of Global Models

The simplicity of formulation and the resulting low computational demand and

numerical stability of global models are attractive features. Implementation of these

models in existing nonlinear dynamic analysis programs is also relatively straightforward.

However, there are some limitations common to lumped models and distributed models.

The axial force-bending interaction is typically neglected or is described by a yield

surface for the stress resultants and an associated flow rule according to classical

plasticity theory (Taucer et al. 1991). For example, among some plasticity-theory-based

models, Sfakianakis and Fardis (1991a, 1991b) proposed a bounding surface plasticity

model to describe the cyclic biaxial bending of RC sections. However, compared with the

fiber models to be introduced in the following section, global models are generally unable

to describe the axial force-bending interaction of RC columns in a rational way.

The hysteretic model involved in global model is phenomenological model of

member behavior, which is based on limited experimental data and cannot be easily

extended to general loading conditions. As has been pointed out by Meyer et al.(1991),

most models only approximate the effect of gravity loads; the interaction between


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bending moment, shear and axial force is described by empirical rules, which can quickly

become extremely complex and are often valid only for the few cases for which they are

calibrated. Also, the parameters of these models cannot be readily established in many

cases, especially for lumped models as mentioned in section 3.2.1.

Finally, the stress and strain response of the cross sections cannot be directly traced

as can be easily done for fiber models. As a consequence, the measures of damage can

only be defined in such overall variables as rotation or moments, instead of more rational

local variable of strain.

3.3 Semi-local Models: Fiber Models

3.3.1 Literature Survey of Fiber Models

The semi-local model reported herein is different from the global model described

previously in section 3.2 in that the stress-strain responses of the cross sections are

calculated locally from the cyclic constitutive relationships of the materials. Thus, no

phenomenological hysteretic model is needed, and the local stress-strain response history

can be directly traced. Fiber element models for RC/PC members belong to the category

of semi-local models and have been regarded as one of the most promising approaches to

modeling RC member. Since a fiber element model is employed in this research, the

following discussion will be focused on the fiber models specifically.

As flexural behavior of frame elements is primarily governed by the longitudinal

normal stress-strain response of the cross sections of the element, the idea of subdividing


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the cross section into layers or fibers is rather straightforward and has been employed by

many researchers since early 1970s. Warner (1969) proposed the concept of ‘fiber

filaments’ for the analysis of the biaxial moment-curvature relationship of a RC column.

Park et al. (1972) presented a, now classical, layered model combing assumed cyclic

stress-strain relationships of steel and concrete, sectional moment-curvature analysis, and

twice integration of curvature to predict the cyclic load-displacement relation of simply

supported RC beam specimens. It is worth noting that 1 hour of computer (IBM 360/44)

time was required to compute the moment-curvature relation for each of the beam

sections in those early days. However, it was not a finite element approach, and was

limited to statically determinate members such as simply supported beam specimens or

cantilever column specimens. Also, this type of analysis is relatively time consuming

because it requires iteration in determining the position of the neutral axis for each cross

section.

Aktan et al. (1975) proposed the first finite fiber element model and applied this

model to dynamic analysis of RC columns subjected to biaxial earthquake excitations and

constant axial load. The classical stiffness approach with cubic shape functions were

employed. Mark and Roesset (1976) presented a fiber model with the incremental

stiffness approach and extended the application to static and dynamic analysis of the RC

frame specimens tested by Gulkan and Sozen (1971). Some of the numerical instability

problems caused by material and member softening were first identified; however, they

avoided numerical instability by using a fictitious concrete stress-stain curve without

strain softening. Kaba and Mahin (1983) employed the fiber model to study the cyclic

behavior of RC sections and later extended to dynamic analysis of RC columns and

frames (Kaba and Mahin 1984). This model incorporated the concept of force


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interpolation functions, which was first proposed by Mahasuverachai and Powell (1982)

for the inelastic analysis of piping and tubular structures. By assuming linear variation of

the section flexibilities along the member and employing an event-to-event solution

scheme, Kaba and Mahins’ model appeared to be the first flexibility-based finite fiber

element model for RC frames.

Zeris (1986) and Zeris and Mahin (1988) improved the original Kaba and Mahins’

model. They discussed two numerical problems: one at section level and the other at

member level. It was demonstrated that the tangent stiffness Newton-Raphson scheme at

the section level was unable to capture the softening behavior of the section. The

numerical problem at the member level was illustrated by a simple cantilever column

example as replicated in Fig.3.7. When a cantilever is displaced beyond the point of

maximum resistance, section 1 at the fixed base of the column starts softening. In order to

maintain equilibrium, sections 2 through 5 start unloading. This behavior generally

cannot be captured with a standard displacement-based model because of the assumption

of a linear distribution of curvature within the member length. They proposed a

four-phase element state determination procedure that mixed both force and displacement

interpolation functions. Their model showed satisfactory performance and was later

extended to biaxial bending problems (Zeris and Mahin 1991). However, as argued by

Taucer et al. (1991), the element state determination procedure is not theoretically clear

and is derived from ad hoc corrections of the Kaba-Mahin model rather than from a

general theory.


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Ithad been recognized by late 80’s (e.g. Meyer et al. 1991) that the flexibility-based

or forced-based formulation could be a remedy to the numerical problems induced by

softening. However, the determination of the element resisting forces, which is often

referred to as element state determination, is not so straightforward for force-based

formulation as for classical stiffness approach. The element state determination

procedures proposed by Zeris and Mahin can also lead to numerical problems because

compatibility is not guaranteed.

This problem had actually been solved in a paper authored by Ciampi and Carlesimo

(1986), in which a flexibility-based beam element formulation with sophisticated local

hysteresis models was presented. They proposed a set of procedures that requires

iterations at the element level and implemented the procedures in a general

displacement-based FEM program (ANSR) for the first time. This approach was later

adopted and further clarified by Taucer et al. (1991), Spacone et al. (1996a, 1996b,

1996c), and Petrangeli and Ciampi (1997) and was applied to the implementation of fiber

beam-column element for RC members. Through an apparently more cumbersome

approach (Petrangeli and Ciampi 1997), this method has proved to be a much more stable

and robust algorithm than the previous methods. Neuenhofer and Filippou (1997)

proposed a modified version of this method, by which the iterations at the element level

Fig. 3.7 Illustration of numerical problems at member level (Zeris and Mahin 1988).


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can be circumvented. Neuenhofer and Filippou (1998) later extended the modified

method to include geometrically nonlinear behavior. Most recently, Coleman and

Spacone (2000) discussed the sensitivity of this method to the number of integration

points or the so-called localization issues; they proposed a regularization technique for

softening sections to improve the problems of loss of objectivity in both the section

moment-curvature response and in the element force-displacement response. These

studies have generally demonstrated the superiority of the approach on account of its

robustness in the presence of strength softening and the lower number of model degrees

of freedom for comparable accuracy in global and local response (Neuenhofer and

Filippou 1998). Therefore, this approach is employed in this research. The procedures of

this approach are to be described in details in Chapter 4.

3.3.2 Displacement-based Formulation Vs. Force-based Formulation

For most of the finite element programs, the displacement-based approach is

employed. Typically, variations of the well-known Newton-Raphson iterative procedure

are used for non-linear FE analysis (Cook et al. 1989, Crisfield 1991). The

Newton-Raphson (N-R) iterative procedure is illustrated in Fig.3.8. With the N-R method,

the standard procedures of displacement-based FE analysis are listed in Table 3.2.

Fig. 3.8 Illustration of Newton-Raphson procedure (Cook et al. 1989).


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Table 3.2 Procedures of displacement-based FE analysis

(1) Iteratively ((Newton-Raphson) solve the system of equations for the incremental

nodal displacement Δui

(2) For each element, all the section deformation increments are approximated from

Δui by using the displacement interpolation function (shape function), a(x)

Δdi(x) = a(x)Δui (3.2)

and the current section deformation vector is

di(x) = di-1(x) +Δdi(x) (3.3)

(3) From section deformation di(x) and the section force-deformation relation (section

constitutive relations), the section stiffness kti(x) and the section force vector D(x)

can be obtained.

(4) Element stiffness and element resisting forces are calculated from integrating kti(x)

and D(x) along the member length:

dxxxxL

it

Tit akaK

0

)( (3.4)

dxxxxL

TiR aDaP

0

)( (3.5)

(5) Assemble the elements’ stiffness to get the global stiffness Kti

(6) Proceed to the next (Newton-Raphson) iteration.


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In the displacement-based approach, the section deformations are approximated

from the nodal displacements by using the displacement interpolation function (or shape

function) as indicated by Eq.(3.2).The cubic Hermitian polynomials are often adopted as

the displacement interpolation function, as cubic interpolation functions produce exact

solution for elastic frame elements. Cubic displacement interpolation function, however,

would result in linear curvature. The curvature of a RC member is highly nonlinear

especially after the maximum resistance has been reached, as shown in Fig.3.7.

Consequently, the adoption of cubic Hermitian displacement interpolation function

generally does not maintain equilibrium along the member and a finer element mesh

would be needed to avoid numerical instability.

In contrast to the inevitable inaccuracy of the displacement interpolation function,

the distribution of the section forces along the member is rather simple. For example,

when there are only nodal forces, the bending moment is linear and the axial force is

constant along the member. Thus, for a member as shown in Fig.3.9, the section force

vector can be expressed in terms of nodal force vector by

PbD xx , (3.6)

where section force vector

xMxM

xNx

y

zD , (3.7)

nodal force vector

yj

yi

zj

zi

MMMMN

P , (3.8)


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force interpolation function

Lx

Lx

Lx

Lx

x

1000

0010

00001

b , (3.9)

and N(x), Mz(x), and My(x) are the axial force and bending moments at section x .

The force interpolation function b(x) is not only simple, it is exact. Thus, equilibrium

along the member is always satisfied in a strict sense. However, the calculation of the

element resisting forces in the force-based approach is not so straightforward as that in

the displacement-based approach (Eq.(3.5)). The procedure proposed by Ciampi and

Carlesimo, as mentioned in section 3.3.1, has to be used to calculate the element resisting

forces. This procedure will be described in details in Chapter 4.

x

y

z

Fig. 3.9 Element local coordinates.

node inode j

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5-1 FE Analysis of RC Frames