slender columns by fem

7/30/2019 Slender Columns by FEM

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The 3rd

ACF International Conference ACF/VCA 2008

706

B.13

NONLINEAR FINITE ELEMENT ANALYSIS OF SLENDER

REINFORCED CONCRETE COLUMNS

Ju Hyoun Cheon- PhD Candidate; Jae Geun Park - PhD Scholar; Moon Young Kim- Professor;

Hyun Mock Shin- Professor; Jun-Hee, Lee - PhD Scholar and executive director of Saman Corporation

Sungkyunkwan University, Suwon, Korea

ABSTRACT: The ACI building code and other's several design codes permit the use of a

moment magnifier approach for the design of slender reinforced concrete columns based on

first-order elastic analysis results. This method is separated in two cases like that structural is

restricted to the side sway and permitted to the side-sway. The design codes stated that

proposed effective flexural stiffness has to be used in first-order elastic analysis and for the

calculation of critical buckling load. Conservative of the results was verified in many

researches for the case. These design codes permit the second-order analysis results directly

in practical design, too. A nonlinear finite element analysis program named RCAHEST

(Reinforced Concrete Analysis in Higher Evaluation System Technology) was used in order to

evaluate the reality second-order behavior of the slender reinforced concrete columns.

KEYWORDS: Mixed Formulation, Flexibility Method, Nonlinear Material Model, Nonlinear

Geometric Analysis, RCAHEST, Slender Reinforced Concrete Columns

1. INTRODUCTION

In the recent years the increased use of slender components in building and bridges has made

it necessary to pay more attention to stability. The strength of a slender column is affected by

many factors such as column length, end-restraint conditions, distribution of bending moment,level of axial thrust, creep of concrete and bracing condition of the column. Most of the recent

analysis methods for slender reinforced concrete columns are based on equilibrium,compatibility, and material properties at the mid-height critical section or at various sectionsalong the column. Kong et al. (1986) proposed a computer-aided analytical method by using

the moment-deflection curves at column mid-height to predict the failure load of a slender

concrete column. Rangan (1990) proposed a method based on simplified stability analysis to

calculate the failure load of slender concrete columns. In the simplified analysis, the moment-curvature relation at column mid-height was converted to a moment-deflection curve for an

axial load value. Bazant et al. (1991) proposed a new analysis method to compute interaction

diagrams for slender columns. Several studies have emerged for the analysis of slenderreinforced concrete columns. Most of them, however, have adopted analytical methods which

assume the deflection curve of the column as a cosine wave or sine wave etc., and solve the

governing differential equations for a column. As the load is increased beyond the elastic limit,


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the deflection shape of the concrete column gradually differs from the deflection curves which

are assumed in the analytical methods, and the application of analytical methods is almostimpossible when the columns are a part of a complex structure. In this study, nonlinear FEM

method is proposed to evaluate the second-order behavior of slender reinforced concrete

columns

2. FORMULATION OF BEAM-COLUMN ELEMENT

2.1 Element Forces and DeformationThe beam finite element without rigid body modes and the element forces and the

corresponding deformations is schematically shown in Figure 1. The element has five

deformation degree of freedom (q1-q5) and corresponding element forces (Q1-Q5). Element

forces and deformations are grouped in the following vectors, respectively:

( ), ( )z z

M x x

( ), ( )N x x

( ), ( )y y

M x x

1 1,Q q

3 3,Q q

4 4,Q q

2 2,Q q 5 5,Q q

Figure 1- Beam element without rigid body modes in local reference system

{ }1 2 3 4 5T

Q Q Q Q Q Q=

{ }1 2 3 4 5T

q q q q q q=

----------------------------------------------- (1)

2.2Beam Element Formulation

The beam element formulation follows the two-field mixed method which uses the integral

form of equilibrium and section force-deformation relations to derive the matrix relationbetween elements generalized forces and corresponding deformation.

( ) ( ) ( )D x b x Q x = ( ) ( ) ( )d x a x q x = ----------------------------------------------- (2)

Where a(x)= deformation interpolation matrix, b(x)= force interpolation matrix

The section force-displacement relation is linearized about the present state and an iterativealgorithm is used to satisfy the nonlinear section force-deformation relation within therequired tolerance. In the mixed method formulation the integral forms of equilibrium and

section force-deformation relations are expressed first. The weighted integral form of the

linearized section force-displacement relation is :

0( )[ ( ) ( ) ( )] 0

LT

D x d x f x D x dx = ----------------------------------------------- (3)

2.3 State Determination

The process of finding the resisting forces that correspond to the given displacements is

known as state determination. The state determination process is made up of two nested


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phase: the element state determination and the structure state determination. These are then

compared with the total applied loads and the difference, if any, yields the unbalanced forcevector. In the Newton-Raphson algorithm the unbalanced forces are then applied to the

structure until external loads and internal resisting forces agree within a specified tolerance.

2.4 Geometrically Nonlinear Flexibility-Based Element State Determination

The classical method in the formulation of frame elements for geometrically nonlinear

analysis derives the geometric stiffness directly from the governing differential equation ofthe second-order theory for bending with axial force (Chen and Liu 1991). Commonly, the

formulation of frame elements for geometrically nonlinear structures is based on appropriate

interpolation functions for the transverse and axial displacements of the member.

Equilibrium in the deformed configuration is satisfied by the force interpolation functions,which in a geometrically nonlinear setting need to include the transverse displacements.

3. REINFORCED CONCRETE FIBER BEAM-COLUMN ELEMENT

3.1 Model AssumptionsThe formulation of the fiber beam-column element is based on the assumption of linear

geometry. Plane sections remain plane and normal to the longitudinal axis during the elementdeformation history. The models used in this study are those discussed in Filippou et al.

(1996).

3.2 Forces and DeformationThe fiber beam-column element is shown in Figure 2 in the local reference system x,y,z. It is

divided into a discrete number of cross sections. These are located at the control points of the

numerical integration scheme used in the element formulation. In this study the Gauss-Lobato

integration scheme is used, since it allows for two integration points to coincide with the endsections of the elements, where significant inelastic deformations typically take place. Each

section is subdivided into ( )n x fibers. The generalized element forces and deformations and

the corresponding section forces are grouped in the following vectors:

Figure 2- Beam-column element in the local reference system : subdivision of cross section into fibers

( )

( ) ( )

( )

z

y

M x

D x M x

N x

=

,

( )

( ) ( )

( )

z

y

x

d x x

x

=

,

1 1 1( , , )

...

( ) ( , , )

...

( , , )

ifib ifib ifib

n n n

x y z

e x x y z

x y z

=

,

1 1 1( , , )

...

( ) ( , , )

...

( , , )

ifib ifib ifib

n n n

x y z

E x x y z

x y z

=

---(3)


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Where D(x)= Section force vector, d(x)= Section deformation vector

E(x)= Fiber stress vector, e(x)= Fiber strian vector

3.3 Fiber Constitutive Models

The nonlinear behavior of the proposed fiber beam-column element derives entirely from the

nonlinear behavior of the fibers. Thus, the validity of the analytical results depends on theaccuracy of the fiber material models. It is important to note that both stress-strain models are

explicit functions of strain. This is a significant feature of the material models in connection

with the fiber model, where fiber strains are determined from section deformations. The stress

determination only involves a function evaluation based in the current fiber stress and strain

and the given strain increment.

4- NONLINEAR FINITE ELEMENT ANALYSIS PROGRAM

4.1 Nonlinear Material Models

The widely used elastoplastic and fracture model for the biaxial state of stress proposed by

Maekawa and Okamura (1983) is used as the constitutive equation for the uncracked

concrete(Fig. 3). For cracked concrete, to consider the tension stiffness effect because of the

bond effect between the concrete and the reinforcing bars (Okamura et al. 1985) and to

describe the compressive behavior of concrete struts between cracks was used in the direction

normal to crack plane and in the direction of the crack plane, respectively. The shear transfer

model based on the contact surface density function (Li and Maekawa 1988) is used to

consider the effect of shear stress transfer due to the aggregate interlock at the crack surface.

These models are shown in the Figure 3 and Figure 4, respectively.

Figure 3- Elasto-Plastic and Figure 4- Equivalent stress equivalent Figure 5- Shear transfer model

strain relationship reversed cyclic loading

The rebar model is developed by applying the equilibrium of force and compatibility of

strain to the stress distribution assumed as sine function between two cracks and was shown

in the Figure 6 and Figure 7. To consider the confining effects was used Mander's confining

effect model like the Figure 8.

1

10

Equivalentstress

Envelope

E K

Eo

Eooo o

=,

:Constant

p

e

max

E Koo

maxp

E K oo= -( )

Equivalent strain


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Figure 6-Yield condition for Figure 7- Reinforcement model for Figure 8- Model for

confined concrete reinforced concrete reversed cyclic loading

4.2 Nonlinear Finite Element Program (RCAHEST)

The proposed structural element library RCAHEST is built around the finite element

analysis program shell named FEAP, developed by Taylor(Taylor, R. L. 2000). RCAHEST

was developed by Kim and Shin(Kim, T. H. et al 2003a, b), at the Department of Civil and

Environmental Engineering, Sungkyunkwan University. The goal of the development is to

apply the program for modeling of various concrete structures under a variety of loading

conditions. Element library of the RCAHEST is shown in Figure 9.

Figure 9- Element library RCAHEST

5- NUMERICAL EXAMPLE

The applicability of the nonlinear analysis program RCAHEST that considers the geometric

nonlinearity was performed. In this case, geometric nonlinearity was considered but didnt

consider the material nonlinearity. The properties and geometric shape of the members are

like that(fig 10). The results show a good agreement with the exact solution.

Figure 10- Geometric details of the specimens

Cf

C

CE

secE

co co2 sp cc cu

'

ccf

'

cof

t

'

tf


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0 0.2 0.4 0.6 0.8 1

Displacement (m)

0

1

2

3

4

5

Load(MN)

Exact

Element, 8 used

Element, 16 used

0 1 2 3 4

PL2/EI

0

0.2

0.4

0.6

0.8

v/L

Exact Soluction

Element 8, Used

Element 16, Used

0 0.2 0.4 0.6 0.8 1

Displacement (m)

0

0.2

0.4

0.6

0.8

1

P

/P

cr

Exact Solution

Element 8, Used

Element 16, Used

Figure 11- Results of the geometric nonlinearity

6- NUMERICAL EXAMPLE

The applicability and adequacy of the nonlinear analysis program named RCAHEST that

consider material nonlinearity and geometric nonlinearity was carried out with reliable tests(J.

K. Kim etc al.). The main purpose of this test was to investigate the effects of the concrete

strength and longitudinal steel ratio on the ultimate load and the behavior of reinforced

concrete columns with the same section and boundary conditions at the ends. In the test

program, three factors were taken : the concrete strength, the longitudinal steel ratio and the

slenderness ratio. In this study, choose only 2 specimens to verify the applicability and

adequacy. The lateral deflections were measured at the mid-height. Material properties and

results of test and analysis presented in Table 1 and Figure 13, respectively. The results are

showing a good agreement with the exact solution (fig. 13).

Table 1- Material properties and results of test and analysis

specimens'

cf (MPa) (%)

Lateral deflection at

ultimate load (mm)

Pu, test

(MPa)

Pu, analysis

(MPa)

60L2-1 14.88 63.7

60L2-160

16.20 65.767.58

100L2-1 29.84 38.2

100L2-1

25.5

100

1.98

32.72 35.0

37.72


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Figure 12- Geometric details of the specimens and subdivision of cross section into fibers

0 10 20 30 40

Laterial Displacement (mm)

0

20

40

60

80

AxialLoad(kN)

60L2-1

60L2-2

Analysis

0 10 20 30 40

Lateral Displacement (mm)

0

10

20

30

40

AxialLoad

(kN)

100L2-1

100L2-2

Analysis

Figure 13- Load and deflection of test and analysis

7- CONCLUSIONS

Most studies to date concerned with the nonlinear analysis of reinforced-concrete frame

structures are based on finite element models which are derived with the stiffness method.

Recent studies have demonstrated the advantages of models derived with the force method.

Flexibility-based element there are no deformation interpolation functions to relate the

deformation along the element to the end displacements, the state determination is not

straightforward and is not well developed in flexibility-based models proposed to date. In this

study, mixed formulation and flexibility-based fiber element method was used that was

developed by F.C. Filippou et al. The nonlinear finite element program RCAHEST that based

on these formulations with applying material and geometric nonlinearities was developed.

The applicability and adequacy was verified with experiments and conclusions are like

followings :

1.Reasonable estimation for the geometric nonlinearities is possible with the nonlinear

analysis program RCAHEST.


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2.Slender reinforced concrete columnss reasonable estimation is possible with

considering material nonlinearity and geometric nonlinearity.

3.Need more effort to prepare reasonalbe estimation method to consider the second-

order effects of slender reinforced concrete columns.

REFERENCE

1. Ansgar Neuenhofer and Filip C. Filippou. (1998), Geometrically Nonlinear Flexibility-

Based Frame Finite Element, Journal of Structural Engineering, Vol. 124, pp. 704-711.

2. B. Vijaya Rangan. (1990), Strength of Reinforced Concrete Slender Columns, ACI

Structural Journal, Vol. 87, 32-38.

3. Spacone., V. Ciampi and F.C. Filippou. (1996), Mixed Formulation of Nonlinear Beam

Finite Element , Computers & Structures, Vol.58, 71-83

4. E. Spancone., F. C. Filippou., and Fabio F. Taucer. (1996), Fiber Beam-Column Model forNonlinear Analysis of R/C Frames: Part. Fromulation, Earthquake Engineering and

Structural Dynamic, Vol.25, 727-742.

5. E. Spancone., F. C. Filippou., and Fabio F. Taucer. (1996), Beam-Column Model for

Nonlinear Analysis of R/C Frames: Part . Application, Earthquake Engineering and

Structural Dynamic, Vol.25, 711-725.

6. Jin-Keun Kim and Joo-Kyoung Yang. (1995), Buckling behavior of slender high-strength

concrete columns, Engineering Structures, Vol.17, 39-51.

7. Kim, T. H., Lee, K. M., Chung, Y. S., and Shin, H. M. (2005), Seismic Damage Assessment

of Reinforced Concrete Bridge Columns, Engineering Structures, Vol.27, 576-592.

8. M. A. Diaz and J. M. Roesset. (1987), Evaluation of Approximate Slenderness Proceduresfor Nonlinear Analysis of Concrete Frames, ACI Structural Journal, Vol.84, 139-148.

9. S. A. Mirza. (1990), Flexural Stiffness of Rectangular Reinforced Concrete Columns, ACI

Structural Journal, Vol.87, 425-435.

10. Taylor, R. L. (2000). FEAP - A Finite Element Analysis Program, Version 7.2. Users

Manual. 1, 2.

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