24

CHAPTER 2

LITERATURE REVIEW

2.1 INTRODUCTION

Concrete is a non-homogeneous material and behaves elastically

over a small load range initially. Inclusion of reinforcement brings in

additional aspects of non-homogeneity because of the involvement of two

materials. Any realistic analysis of reinforced concrete structures, should take

the effect of these non-homogeneities which reflect in several complexities

like cracking, nonlinear material behaviour, the loss of bond between concrete

and steel etc. Concrete by nature cracks at a very low tensile stress and hence

cracking of concrete is a major nonlinearity. The other nonlinearities

connected with prestressed concrete are nonlinear stress-strain law,

compression softening of concrete, creep, shrinkage, bond-slip etc. The

nonlinear analysis and design procedures are so complex that their use by

structural designers is virtually impossible. Nonlinear design techniques tend

to be interactive and computer oriented. The spectacular developments in

matrix, mathematical programming and computer techniques have led to

development of nonlinear finite element models for full range analysis of

concrete structures.

The displacement based finite element method endowed with

smartness, generality in solving complex structural problems, manifested

itself as the broadest yet most accurate tool of analysis. Finite element models

have been developed for material nonlinearity of reinforced concrete

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structures including time dependent effects due to load history, creep,

shrinkage and aging of concrete. Nonlinear models if properly generated lead

to better and economical structural design. Though some research has already

been taken place on analysis of box girders, no special attention has been paid

to the nonlinear finite element analysis of prestressed concrete box girders.

This chapter discusses the contributions in the field of finite element analysis

of reinforced and prestressed concrete box girders.

There are various analytical methods available for analysis of box

girders. However, these analytical solutions are limited in scope and are not

applicable to arbitrary shapes, conditions, irregular stiffening, various support

conditions, cut outs etc. Early box girder designs employed wall thickness

large enough to render distortional and warping effects negligible, so that

simple beam theory and St. Venant’s torsion theory are adequate analytical

tools. However, with the present tendency to have more slender sections to

reduce the self weight and prestress, distortional and warping effects may

need to be considered. Finite Element method due to its generality emerges as

most potent technique for analyzing the box girder bridges.

2.2 FINITE ELEMENT ANALYSIS OF BOX GIRDERS

Originally box girder bridges were studied using different types of

finite elements other than shell elements. Attempts were made to achieve

maximum compatibility between the displacements of adjoining elements. In

the finite element method, a box girder is divided into elements in

longitudinal direction also as shown in Figure 2.1, and hence there is an

advantage of possibility of analyzing problems with variation of geometry,

thickness, material properties of loads, along the length of the bridge. Also

any shape of boundary and support conditions can be entertained at bridge

ends.

26

Figure 2.1 Discretization of single cell box girder

The earliest attempt in this direction is that of Aneja and Rolls

(1971). They analyzed a unicellular curved concrete box girder bridge with

vertical webs. Flanges, deck slab and soffit slab were discretized with flat

plate elements with curved boundaries and webs with cylindrically curved

rectangular elements.

2.3 FINITE ELEMENTS USED IN ANALYSIS OF BOX

GIRDERS

Box girders are conceived as thin shell structures comprising of

folded plates monolithically connected at joints. Some of the elements used

are discussed in the further sections.

27

2.3.1 Flat plate elements

This element was developed by superposing bending and stretching

behaviour. The structure was modeled by assembling flat elements located

with their vertices lying in the middle surface of the shell. The element is

oriented properly by applying coordinate transformation in calculating the

stiffness matrix and load vector.

An early attempt in using this element was made by Mehrotra,

Mufti and Redwood (1969). They used flat triangular elements as shown in

Figure 2.2, formed by combining uncoupled in-plane and bending actions.

Linear in plane and cubic normal displacements were assumed, resulting in a

15 degrees of freedom element. Six degrees of freedom were used at each

node out of which rotation in the z-direction and the corresponding stiffness

was taken as zero. This caused singularity when all elements at a joint meet in

a plane. No remedy was used for the ill-conditioning occurring where

elements meet nearly in a plane but are not exactly coplanar. A single box

right ended straight bridge with vertical webs and cantilever flanges was

analyzed and results were compared with the experimental values.

x

z

z

y

vx

uy

v'In plane' forces and deformations

'Bending forces' and deformations

Figure 2.2 Flat elements subjected to inplane and bending actions

28

Sisodiya, Cheung and Ghahi (1970) used parallelogram elements

for webs and in-plane triangular elements for deck and soffit slabs and

analyzed single cell skew box girder bridges to study the effect of element

sizes and found that for curved box girder bridges finer meshes with elements

of aspect ratio 1:2 gave better results than coarse discretization (aspect

ratio 1:4). In this analysis, curved sides of flanges were approximated by

straight edges of parallelograms. For curved webs, flat rectangular elements

were used. Single span and continuous two span bridges were analyzed.

Results obtained were compared with experimental results from models. Also

comparison of parallelogram and triangular elements was made. Results in

both the cases were almost identical, but less computer time was required for

triangular elements.

Chu and Pinjarkar (1971) analyzed simply supported curved box

girder bridges with out intermediate diaphragms by direct stiffness element

formulation. The deck and soffit slab were discretized with horizontal circular

sector plate elements. The solution was exact and economic with limitations

like not being applicable to arbitrary geometry, sloping deck, inclined webs,

orthotropic material properties.

The flat plate element accepts rigid body motion with out strain in

spite of certain definite limitations (Ashwell, 1976). These include

(a) exclusion of coupling of stretching and bending within elements. (b) The

difficulty of treating junctions where all the elements are coplanar due to the

presence of a null stiffness matrix corresponding to rotation about the axis

normal to the plane. (c) The presence of discontinuity bending moments,

which do not appear in continuously curved actual structure at the element

juncture lines.

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2.3.2 Curved elements

The following are the requirements for an efficient numerical

analysis of box girder bridges as stated by Sisodaya, Cheung and Ghahi

(1972).

The method should be capable of analyzing any box girder

bridge having arbitrary plan, variable cross section and any

support conditions.

A box girder bridge can be analyzed by thin shell type

elements. All the six degrees of freedom u, v, w, x, y, z

must be represented. Suitable bending elements with nodal

parameters w, x and y are available and hence the inplane

elements should include u, v, z. An element with more nodal

parameters or mid side nodes is not recommended, because

such elements increases the band width of the stiffness matrix

and hence the cost of computation.

The chosen plane stress elements should result in small

number of equations and small bandwidth.

These requirements were satisfied by shell elements of

parallelogram or quadrilateral shapes. The shell elements were obtained by

combining the in-plane stiffness of a parallelogram element and two

triangular elements respectively. These shell elements are applicable for the

analysis of box bridges of arbitrary plan and variable depth. Analysis of single

cell skew and curved box girder bridge with span divided at only six equally

spaced sections, showed that these elements give accurate results.

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Fam and Turkstra (1975) developed a finite element program using

plate bending elements with four corner nodes for flanges, cylindrical in-

plane and bending elements for webs. A new conforming annular plane stress

element with corner nodes was developed to reflect appropriate stress

variations in the flange-web junction. Diaphragms were discretized with the

rectangular 8 degrees of freedom in plane element reported by Przemienieck

(1968). Near the box corners, the mesh divisions were relatively small to

produce accurate results in the regions of high stresses and steep stress

gradients. This analysis incorporates coupled in-plane and bending actions

and anisotropic properties. The displacements, u and v along the x and y

directions, were determined using the rectangular or annular flange elements

with 8-term polynomials, linear in y and cubic in x. For w, a 12 term

polynomial quadratic in y and cubic in x was used. As the local and global

axis coincide in case of flange elements, no transformations were necessary.

In web elements, the polynomials for u contains two extra terms,

corresponding to two mid side nodes taken on transverse edges of the

rectangular or conical elements as shown in Figure 2.3. These extra nodes,

with the only displacement u, were taken to ensure a better representation of

rapidly varying longitudinal strains along the depth of the webs.

Figure 2.3 Rectangular and conical web elements

z,w

y,v 3

2

Z,W5

4

x,uX ,U

1 Y ,V

6

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Transformation matrices for these elements were in terms of , the

inclination of slant web or of cone-generator with horizontal. The author also

used triangular and rectangular diaphragm elements, which were assumed to

have only in-plane stresses. These were 6 and 8 degrees of freedom elements,

with bilinear variations of displacements u and v. Curved slab bridge and twin

cell curved box bridge problems were solved for concentrated loadings.

Results were compared with finite strip solution of Scordelis and curved

folded plate solution of Chu and Pinjarkar (1971). The results agreed well

with the analytical solutions, and were found to be much superior to the finite

strip solution.

Moffatt and Lim (1977) discussed various flexural elements used

for box girder bridges by other researchers i.e., elements such as quadrilateral

element with 2 in-plane and 3 bending degrees of freedom at each node and

also parallelogram elements. They stated the need for higher order elements

and also pointed out that combination of triangular and rectangular elements

for idealization of skew bridges gave more accurate results than parallelogram

elements. These elements and their combinations are applied to a three cell

straight box girder bridge, a curved single cell box bridge with projecting top

flange and a skew straight composite twin-cell box girders. All these results

were compared with experimental results obtained on Perspex models. It was

found that the quadrilateral element with 4 in-plane and 3 bending degrees of

freedom was more efficient than others and gave acceptable results. This

element was more useful for regions of high stress-gradients such as those in

shear lag phenomenon.

2.3.3 Isoparametric elements

For the analysis of thick-walled box girder bridge, 3-D elements are

preferable. Zienkiewicz and Too (1972) utilized semi-analytical approach to

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represent the structure with 3-D prismatic elements, by discretising the cross

section of bridge with 2-D elements and using Fourier series expansion along

length. As such the method is useful for straight or curved prismatic bridges

only and with only simple supports. This system is demonstrated by applying

to straight and curved thick bridge boxes.

Vanzyl and Scordelis (1979) analyzed curved prestressed concrete

segmental box girder bridges using direct stiffness method. They used the

skew-ended finite box elements with 8 degrees of freedom at each of its two

end nodes. The element was superior than beam elements but lacked the

accuracy of plate bending element.

2.3.4 Degenerated shell elements

After the introduction of numerical integration by Irons (1966), 2-D

and 3-D elements found a wide scope in analysis. These elements had the

novel property of fitting into any arbitrary shape. Also, due to the use of shape

functions, element formulation and computer coding became easier. But, in

the application of these 3-D elements to shells, large number of unnecessary

unknowns appeared and the analysis became cumbersome. Then, emerged the

Ahmad’s ‘membrane stack element’ (Ahmad, 1970). This element was

“degenerated” or “degraded” from a 20 noded 3-D element with a total of 60

degree of freedom to obtain a 8 noded element with 40 degree of freedom.

Only 8 nodes (corner and midside) on the mid surface of the element were

retained as shown in Figure 2.4. Three displacements u, v, w and two

transverse rotations x and y were taken as unknowns at each node, thus

giving 40 degrees of freedom in total. The element was assumed to be made

up of a stack of thin membranes, connected by eight rigid stalks with zero

transverse strain and constant transverse shear along the thickness.

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Figure 2.4 A 20-Node solid element degenerated to 8-node element

The error caused by constant shear is corrected by a shear

correction factor. However, Ahmad’s element did not yield satisfactory

results for thin shells. Then, the element was modified by adopting reduced

integration (Zienkiewicz, 1971). Reduced integration introduces a defect

known as spurious mode. A spurious mode is one in which the displacement

mode of the element offers no resistance to the applied loads. This results that

the stiffness matrix is not stiffened by the spurious shear strain because the

spurious bending modes produce zero shear strain. To avoid the defects due to

spurious shear strains, further modifications were suggested by adopting,

selective integration (Pawsey, 1971) to obtain proper integration of energies

other than shear avoiding spurious shear, addition of a bubble function (Cook,

1973) to include biquadratic mode of displacement, elimination of certain

variables at mid-side nodes (Irons, 1973) to give the element edge a thin beam

effect and to reduce unknowns etc. Analytical investigations reported that

adding an internal node improved the performance of an element (Cook,

1989).

A four nodded Bilinear Degenerated Shell (BDS) element with six

degrees of freedom at each node was also used. The vertical deflection and

normal stresses in both flanges were plotted and compared with results from

both experimental investigations and higher order finite element solutions.

ζ

η ξ

ζ

η ξ

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This element coupled with a reduced integration technique, performs

accurately in both thick and thin shell situations.

2.3.5 SemiLoof shell elements

Semiloof shell elements originally conceived by Loof (1966) were

the most versatile yet highly complex, isoparametric on conforming direct

stiffness, element capable of accepting exact rigid body motions with out

undergoing strain. This element can model sharp corners, multiple junctions

using higher order exact solution with in the element with minimum effect of

degree of nonconformity between neighbouring elements. The specialty of

this element is that the nodes are selected along the edges corresponding to

Gaussian points (2 X 2) in addition to conventional corner, mid side and

central nodes. The nodal configuration of a quadrilateral SemiLoof element is

shown in Figure 2.5. To succeed in the quadrilateral patch test, an extra initial

degree of freedom using the bubble function is introduced.

Figure 2.5 SemiLoof nodal configuration

Irons (1974) has published this efficient super parametric thin shell

element which contained many important sophistications. The most important

of these being the positioning of rotation variables at Gauss points on edges

instead of at corners. Due to these, the element proved very efficient for

35

situations where two or more thin shell elements meet at angles such as in box

girders.

Irons and Ahmed (1980) explained in length the formulation and

coding of SemiLoof element, bringing out the importance of its various

special features.

Javaherian, Dowling and Lyons (1980), used this element for

nonlinear analysis with some modifications.

2.4 NONLINEAR FINITE ELEMENT ANALYSIS OF

REINFORCED CONCRETE STRUCTURES

During the two decades considerable amount of research effort was

devoted to the development of numerical models for the nonlinear behavior of

reinforced concrete structures. Most of these numerical models were

conceived by coupling a displacement formulation of the finite element model

together with a set of partial constitutive models for the aspects of nonlinear

behavior such as the stress-strain relationship, the initiation and propagation

of cracks, bond between steel and concrete, and time dependent phenomenon

such as creep and shrinkage of concrete. Recently, the prestressing effects

have been introduced as an extension to the existing models. The prevalent

finite element models, which are adopted for various prestressed concrete

shells, are discussed below.

Hand, Pecknold and Schnobrich (1973) presented a layered

nonlinear finite element analysis including the elasto-palstic behaviour of

steel, bilinear-elasto-plastic behaviour of concrete and tension stiffening of

concrete. They used the incremental variable elasticity technique to obtain the

load deflection curve for any general plate or shell. They also demonstrated

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the need for a shear retention factor to provide the torsional and shear

stiffness for cracked concrete.

Lin and Scordelis (1975) in their paper presented a nonlinear finite

element analysis of RC shells of general form. The analysis was done to trace

load-deflection response, the crack-propagation through the elastic, in-elastic,

and ultimate load ranges. They adopted a layered approach with an

incremental tangent stiffness solution technique. They found in their study

that the tension stiffening effect of concrete between cracks had a significant

influence in post-cracking load-deflection response of under reinforced

concrete structures. The importance of failure criteria was also investigated.

Scordelis (1989) gave a brief review of research in development of

nonlinear analytical models, methods of analysis and computer programs to

trace the structural behaviour of reinforced concrete shells under increasing

loads through their elastic, cracking, inelastic and ultimate ranges.

Crisfield and Wills (1989) in their paper described the application

of a series of different concrete models to the analysis of a number of

reinforced concrete panels tested by Vecchio and Collins at the university of

Toronto.

Roca and Mari (1993) presented a general formulation for the

nonlinear, material and geometric instantaneous and long-term nonlinear

analysis of prestressed concrete structures. The formulation is based on

discrete treatment of the prestressing tendons, where both the prestress

geometric and mechanic efforts are introduced consistently with the

displacement formulation of the finite element method.

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Duraiswamy (1994) proposed a degenerated nine node, two

dimensional, curved isoparametric element with five degrees of freedom at

each node. He used the layered approach, dividing the concrete and

reinforcing steel into layers. The prestressing force is introduced as work

equivalent nodal forces.

Foster, Budino and Gilbert (1996) formulated a rotating crack finite

element model for the analysis of reinforced and prestressed concrete

structures. They used discrete element to model the concrete and steel with

bond slip incorporated into the formulation of the bar element used to model

the steel reinforcement.

Paramasivam et al (1997) have developed layered shell elements

for moderately thick composite shells. The element consists of a number of

bonded layers of orthotropic materials. In the nonlinear analysis, due to

bending, different sandwich layers will be in different states of strain. So, a

layered approach can be adopted assuming the sandwich layer to have

constant strain across its thickness even though the strain varies in the

thickness direction of the whole lamina.

2.5 CONSTITUTIVE LAWS FOR CONCRETE

The information required in any finite element calculations for

reinforced concrete is the multi-dimensional stress-strain relations, which

adequately describe the basic characteristics of reinforced concrete materials

subjected to monotonic and cyclic loading. These are called constitutive

relations and failure theories. Although a large number of material models for

concrete based on different theories have been published with in the last few

decades, there is no generally accepted constitutive law for this material

(Hofstetter, 1996). Investigations in constitutive modeling relevant to the

38

present problem are reviewed and critical review of research on behaviour of

concrete at failure and cracked concrete are reported in the section. Some of

the constitutive models usually adopted in finite element analysis are

presented.

2.5.1 Stress – Strain laws Concrete is a heterogeneous material with a nonlinear stress strain

relationship and its properties are entirely different when in compression and

tension. Its stress strain curve is linear and brittle in tension, but is nonlinear

and ductile in compression. The properties of concrete are dependent on many

factors. In most of the concrete structures, the concrete is subjected to multi

axial state of stress and its behaviour under such a condition becomes more

complicated. There are four main groups of approaches for defining the

complicated stress-strain behaviour of concrete under various stress states.

They may be classified as follows.

1. General representation of stress-strain curves by

curve-fitting methods, interpolation techniques or

mathematical functions

2. Nonlinear elasticity theories, which are further divided as

a. Hyper elastic

b. Cauchy-elastic

c. Hypo elastic models

3. Perfect and work-hardening plasticity theories and

4. Endochronic theory of plasticity

Independent research has been conducted by various authors to

study the behaviour of concrete under uniaxial, biaxial and triaxial loading

conditions. Many investigators have used a simple parabola-rectangle model

for the stress strain curve of concrete.

39

Liu, Nilson and Slate (1972) proposed an orthotropic biaxial-strain

law in matrix form, which accounted for the influence of poisson’s effect and

micro crack confinement effects. The material properties are described in

terms of total strain in the principal stress directions.

Darwin and Pecknold (1977) extended the model of Liu et al (1972)

to incorporate cyclic loading and strain softening in compression and

proposed a stress-strain law designed to be used in conjunction with the finite

element techniques. They used an incremental (tangential) rather than secant

form for the proposed stress-strain law. For biaxial compression they

proposed a family of curves, which use equivalent uniaxial strains.

Ma and May (1986) have used a mathematical model having a

straight initial portion, an ascending parabola and a descending straight line.

Vecchio (1989) mentioned that a secant stiffness approach can be

successful, while being less restrictive on constitutive laws that can be

implemented to the solution procedures required.

Shayanfar (1997) have observed that, the shape and length of the

descending branch of the tensile stress-strain curve of concrete have

significant effects on the computed responses. These parameters are

controlled by the values of ultimate tensile strain and tensile strength of

concrete.

2.5.2 Concrete cracking

The tensile weakness of concrete and the ensuing cracking that

results therefrom, is a major factor contributing to the nonlinear behaviour of

reinforced concrete elements. Before failure, the material has isotropic

40

properties, but beyond cracking, it has orthotropic properties. Steel

reinforcement and/or prestressing steel may be used to provide the necessary

tensile strength. The numerical simulation of cracking is an important aspect

of the finite element analysis of massive structures. A number of early studies

to predict numerically, the behaviour of reinforced concrete structures by

finite element techniques focused on the inclusion of cracking behaviour in

the mathematical model.

The first finite element model of reinforced concrete to include the

effect of cracking was developed by Ngo and Scordelis (1967) who carried

out a linear elastic analysis of beams with predefined crack patterns. The

cracks were modeled by the separation of nodal points.

Nilson (1968) introduced progressive cracking propagating through

the structure by disconnecting or separating elements on each side of a node

when stresses at that node achieved values sufficiently high to produce the

crack. This procedure therefore actually produced a crack by the physical

separation into two sides across a crack. Changing the topology of the

mathematical model was a problem in implementing the procedure. However,

this was overcome by double noding those nodes across which the crack was

to propagate.

Rashid (1969) pioneered later an approach that looked at the effect

of cracking on the overall behaviour of the structure. This model, became

known later as smeared crack approach. In the fixed crack model, when a

crack occurs and remains open, its direction is fixed and does not change

under subsequent loading. This initial crack may close and a secondary crack

is restricted to form orthogonal to the initial crack. This model is usually

called as the fixed crack model. The freezing of directions of the primary and

secondary cracks regardless of the actual orientation of the principal stresses

41

under subsequent loading may result in violation of the cracking criterion. He

has represented the cracked concrete as an elastic orthotropic material with

redundant elastic modulus in the direction normal to the crack plane.

The rotating crack concept in finite element analysis of reinforced

concrete is that, after cracking takes place, the crack direction is always

perpendicular to the direction of the major principal strain axis during the

course of loading. This concept was extended and modified by other

researchers.

Milford and Schnobrich (1985) demonstrated the rotating crack

model, which assumes that the crack direction is always normal to the

direction of the principal tensile stress or strain. Subsequent cracks are not

restricted to form normal to the initial one. The principal concrete strain

directions are allowed to change after the initial cracking, so that secondary

cracks can develop at other directions other than normal. Milford and

Schnobrich in their work have neglected the nonlinearities associated with the

normal shear stress and used a constant shear correction factor of 1.2. They

have demonstrated the application of the rotating crack model to reinforced

concrete panel and slab structures.

Bazant and Gambarova (1980) were the earliest researchers to have

developed a realistic model for modeling shear transfer across the cracks

based on the experimental data. Prior to their work, the shear transfer was

modeled by reducing the shear modulus Gs by a shear retention factor

β (0 ≤ β ≤ 1). They also observed that the phenomenon of shear transfer is

highly nonlinear.

Cervenka (1985) noted that if shear strain occurs on the crack

plane, such as in case of non-isotropic reinforcement or non-proportional

42

loading, the shear resistance of cracks should be included. He maintained that

the tensile strength affects not only the cracking stress but also the entire

stress-strain curve and the failure load.

2.5.3 Tension stiffening effect

Gilbert and Warner (1978) introduced the tension stiffening effect

with reference to steel. The steel stiffness is increased to account for the

tension carried by concrete. The stress in the concrete is exact at the crack

whereas, the steel stress is increased to include the tension in concrete.

Roca, Mari and Scordelis (1989) incorporated tension stiffening

effect by modifying the stress-strain relationship for steel reinforcement.

Fictitious higher modulus is used for reinforcing steel for modeling tension

stiffening. Once concrete cracks, the stress normal to the cracks is zero unlike

other models incorporating tension stiffening effect where the stress in

concrete is gradually reduced to zero.

Chan, Chung and Hyuang (1993) concluded that the accuracy of

nonlinear analysis of reinforced concrete structures is greatly improved by

taking the tension stiffening effect into consideration. In their model, the

tension stiffening effect was dependent on ratio of reinforcement, the strength

of concrete, moduli of concrete and steel etc.

2.6 CONSTITUTIVE RELATION FOR STEEL

Unlike concrete, steel reinforcement is treated as axial bar and only

uniaxial stress-strain curves for steel are considered in the finite element

analysis. In the finite element programming, the stress in steel at various loads

are needed during the analysis. This can be achieved by developing analytical

43

expression for stress-strain curves of steel, which gives the stress in steel for

any strains.

2.6.1 Modeling of normal steel

In the finite element analysis of reinforced concrete structures, the

reinforcement models may be placed into three categories: discrete, smeared

and embeded.

Naaman (1983) has reviewed the various stress-strain relationships

for steel and suggested that the relationship suggested by Menegotta and Pinto

(1973) can be used in nonlinear analysis of concrete structures, since the error

is least in this case compared to others.

Allwood and Bajarwan (1989) described a new method of

representing the steel in finite element analysis of reinforced concrete

structures in which the steel and the concrete are analyzed separately. The

forces between the steel and concrete are used in an iterative method, which

brings the two solutions together. The method converges very rapidly and it is

mentioned that the computational effort is principally dependent on the mesh

chosen for the concrete.

An embedded reinforcement model is used by Rajbaran and

Phippes (1994) in their finite element software DENA. The model presented

by them is quite general in the sense that it can be used for two and three

dimensional concrete elements and reinforcement of arbitrary shape. They

assumed full bonding between the reinforcement and concrete. Moreover,

only longitudinal strain of reinforcement is considered.

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2.6.2 Modeling of prestressing steel

Jirousek, Bouberguig and Saygun (1979), in their work to facilitate

the analysis of prestressed box-bridge problems, presented treatment of

prestressing cable. They assumed a cable traversing a parabolic curve in the

mid-plane of a shell element. The load components were computed and

converted into equivalent nodal loads for the shell elements. If there were

more than one cable segment passing through the shell element, equivalent

nodal loads were obtained for each of these segments and all these were

added to the usual load vector of the element. Using this augmented load

vector, the usual solution procedure was carried out to give the final

displacements and stresses in the box-girder bridge. Due to the assumptions

made, the prestressing takes the form of only an additional load case. Hence,

the final results of analysis of a loaded prestressed bridge will be approximate

to a certain extent. They have demonstrated the analysis of a straight single

cell bridge subjected to prestressing loads only. Some of the results given are

compared with those obtained by the ‘standard beam approach used in

practice’.

Figueiras and Povas (1994) have simulated prestressing steel by

one-dimensional curved elements embedded in the finite elements employed

to discretize the concrete structure. This formulation is to be incorporated

with curved thick shell elements.

Kotsovos and Pavlovic (1994) have described a simple way to

incorporate the constitutive law for prestressing steel by modifying the stress-

strain curve of the prestressing steel. According to their approach, the origin

of the stress-strain curve is to be shifted to the level of prestress in the strand

and the strand can be treated as an unstressed strand.

45

Balakrishnan (1991) has conducted tests on a series of segmental

prestressed concrete beams and also developed a method for analysis of

segmental beams using plane stress rectangular elements. He found that the

joint stiffness coefficients are highly nonlinear and depend on the crack width,

slip, the key angle and the amount of prestress.

Vasudevan (1993) has conducted analytical and experimental

investigations to study the behaviour of precast prestressed segmental box

girder bridges. He used finite element method and modeled prestressing bars

as discrete bars. The analytical results are supported by testing of one 10.0 m

span segmental box girder bridge model. The results revealed that the

segmental girders undergo relatively more deformations than the monolithic

ones.

Aravindan and Rajeeva (1993) tested in the elastic range, two

single cell reinforced concrete box girder models of skew angles 00 and 300

with three intermediate diaphragms and compared the results with the finite

element analysis to study the effect of skew angle and intermediate

diaphragms.

Shih Toh Chang (2004) observed that, if the tendon takes the

parabolic profile, the shear lag effect, caused by dead load plus prestress,

remains the same as the dead load acting alone.

Babu Kurian and Devdas Manoharan (2005) carried out analytical

and experimental investigations to develop simplified equations to predict the

collapse load of concrete box girders. He suggested a set of correction factors

for transverse moments obtained from the simplified frame analysis.

46

Luo, Li and Tang (2002) adopted finite segment method for

analyzing shear lag effects in box girders with an assumption that the

span-wise displacements of the flange plates are described by a third-power

parabolic function. In order to obtain the longitudinal stresses under the shear

lag effect, the element stiffness equations are developed based on the

variational principle by taking the homogeneous solutions of the differential

equations as the displacement functions of the finite segment. The effect of

two major parameters on shear lag is investigated for cantilever and

continuous box girders with varying depth under three kinds of loads. It is

shown that the height ratio, in addition to the flange width to span length

ratio, has a significant influence on the shear lag.

2.7 SUMMARY OF THE LITERATURE SURVEY

A proper description of material behaviour requires the

necessity of obtaining the material data.

There is discrepancy in the failure envelope and constitutive

relations obtained by various research groups.

In most of the nonlinear analysis done for concrete, the

constitutive laws used were based on plastic theory. The

main drawback of this theory is that the unloading in the

stress-strain curve cannot be easily incorporated. The elastic

theory based approach is an easy alternative, which utilizes

the equivalent uniaxial strain concept.

Of the different approaches for defining the complicated

stress-strain behaviour of concrete under various states of

stress, the easiest and the most appealing way is to represent

the stress-strain curve by using curve-fitting methods,

interpolation techniques or mathematical functions.

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The linear-elastic models can be significantly improved by

assuming a nonlinear elastic stress-strain relationship in

secant modulus form.

The classical theory of plasticity, which has been developed

originally for metals, needs some modifications for applying

to concrete.

Tension stiffening play a significant role in reducing post

cracking deformation of reinforced concrete structures and is

an important parameter to be considered in the analysis if the

behaviour of the structure is to be studied beyond cracking.

The shape of the average stress-strain curve of steel bars can

be modeled as a trilinear curve with the slope of the post

yield branch being only a fraction of the slope before

rupture.

Tension stiffening can be modeled by developing two

constitutive laws; one for concrete in tension and another for

steel bars embedded in concrete. Both stress-strain curves

relate the average stresses to the average strains of steel and

concrete.

One of the approaches to model tension stiffening is to relate

the stress-strain relationship to the reinforcement ratio,

strength of concrete and elastic modulus of concrete and

steel.

The cracking strength of concrete varies with the presence of

reinforcement and is different from that of plain concrete.

No attempt has been made to predict the cracking strength of

concrete for different percentages of normal steel.

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The first major crack forms at 90% of the average concrete

tensile strength. Shrinkage cracks and cracks caused by

bleeding around the aggregates clearly do exist prior to

loading the concrete. These initial cracks spread to cause

major cracks of failure. The cracks, which are initially

invisible, become visible with the application of the external

loads and this contributes to the generally obtained nonlinear

stress-strain behaviour.

Discrete crack modeling requires remeshing as crack

propagates. Smeared crack approach neglects the stress-

singularity at the crack tip, and therefore gives an inadequate

prediction of crack. Smeared crack can be used if the

behaviour of concrete is not dependent on a few cracks,

wherein, the discrete crack model is to be used.

Rotating crack models assumes that the crack direction is

always normal to the direction of the principal tensile stress

or strain. Subsequent cracks are not restricted to form

normally to the initial one. Rotating crack model or

swinging crack model will give a collapse load that is less

than or equal to that given by the fixed crack model.

Implementation of nano orthogonal crack model

encountered significant numerical difficulties when state

changes (yielding, crack opening, crack closing etc.)

occurred within an increment. This is also difficult since a

number of cracks have to be stored and monitored at a point.

To analyze reinforced concrete structures in the ultimate

load range, an incremental iterative analysis can be

employed. Initial stiffness eliminates the problem of non-

positive definite stiffness matrices after concrete cracks in

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both the directions and steel yields. It is preferable to set a

displacement criterion for the solution tolerance.

Reinforced concrete can be analyzed by layered approach to

overcome the difficulty of its non-homogeneity. The steel

and concrete can be specified as separate layers with

different material properties. The deficiency observed is that

in the layered approach the concrete between the steel rods

is not considered.

The effect of prestressing force on the stress-strain curve of

concrete in tension in prestressed concrete members is not

properly known.

Equations describing the stress-strain behaviour of

prestressed concrete in tension considering the amount of

normal steel are not available.

Prestressing can be included in the analysis by separating the

material and force effects and shifting the origin of the

stress-strain curve of the prestressing steel.