chapter 2 literature review - shodhganga : a...
TRANSCRIPT
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
25
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.
29
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.
30
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
31
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
32
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.
33
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
34
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
36
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.
37
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.
44
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.
47
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.
48
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
49
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.