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Design procedure for prestressed concrete beams
Article  in  Computers and Concrete · February 2014
DOI: 10.12989/cac.2014.13.2.235
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Computers and Concrete, Vol. 13, No. 2 (2014) 235-253
DOI: http://dx.doi.org/10.12989/cac.2014.13.2.235 235
Copyright © 2014 Techno-Press, Ltd.
http://www.techno-press.org/?journal=cac&subpage=8 ISSN: 1598-8198 (Print), 1598-818X (Online)
Piero Colajannia
, Antonino Recuperob
and Nino Spinella

Dipartimento di Ingegneria Civile, Informatica, Edile, Ambientale e Matematica Applicata,
Università di Messina, Messina, Italy
(Received November 14, 2012, Revised September 3, 2013, Accepted September 16, 2013)
Abstract. The theoretical basis and the main results of a design procedure, which attempts to provide the
optimal layout of ordinary reinforcement in prestressed concrete beams, subjected to bending moment and
shear force are presented. The difficulties encountered in simulating the actual behaviour of prestressed
concrete beam in presence of coupled forces bending moment - shear force are discussed; particular
emphasis is put on plastic models and stress fields approaches. A unified model for reinforced and
prestressed concrete beams under axial force - bending moment - shear force interaction is provided. This
analytical model is validated against both experimental results collected in literature and nonlinear numerical
analyses. Finally, for illustrating the applicability of the proposed procedure, an example of design for a full-
scale prestressed concrete beam is shown.
Keywords: shear, flexure, prestressed concrete, design, plasticity
1. Introduction
In the last forty years, theoretical and experimental investigations have clarified the most
important aspects of shear failure in Reinforced Concrete (RC) element (Nielsen, 1984; Vecchio
and Collins, 1986, Collins et al. 1996; Russo and Puleri 1997, Russo et al. 2004, Russo et al.
2005) and many models to predict the shear and flexural capacity of Prestressed Concrete (PC)
beams are available in literature (Au et al. 2009, Ahn et al. 2010, Au et al. 2011, Gocic and
Sadovic 2012). However, little consensus as developed among researchers as to which is the most
accurate and suited for use in design. Moreover, historically, the many of codes propose two
different models, for RC structures on one side and for PC elements from the other (ACI
Commitee 318, 1983, Eurocode 2 2002).
The procedure to design RC elements subjected to shear (V) and bending moment (M) reflects
the classical truss model approach. The idea to use truss models of following the flow of stresses
goes back to Ritter (1989). Based on careful observations of structure behaviour as well as a
systematic understanding of basic principles of structural mechanics, Mörsch (1908) has greatly
advanced this concept and introduced the classical 45°truss model, whose simplicity and clearness
Corresponding author, Ph.D., E-mail: nino.spinella@unime.it
a
Associate Professor, e-mail: pcolajanni@unime.it
b
Assistant Professor, e-mail: antonino.recupero@unime.it

Piero Colajanni, Antonino Recupero and Nino Spinella
are striking even today.
The application of the theory of plasticity to structural concrete beams and walls started in the
late 1950s. Basic work, in this field, was mainly carried out by two groups of researchers around
Thürlimann (1983) and Nielsen (1984). Their contribution tries to provide an overview on limit
analysis methods for structural concrete beams subjected to shear and bending moment and to give
a structure at the considerable provisions of Eurocode (2002).
Though sophisticated Non-Linear Finite Element Analysis (NLFEA) can be nowadays
performed using accurate models (Vecchio and Collins 1986, Bertagnoli et al. 2011), simplified
mechanical models are still definitely needed, at the aim to speed up the design and the analysis.
In the case of RC elements, by using the truss models and the stress field theory, a model to
evaluate the shear strength in presence of axial force has been proposed and discussed by several
authors, referring to rectangular cross sections (Fanti and Mancini, 1995), to T or I-shaped cross
sections (Recupero et al. 2003), and to circular cross sections (Rossi and Recupero 2013). This
model was also generalized in presence of Fibres Reinforced Concrete (FRC) (Colajanni et al.
2008a, Spinella et al. 2010, Colajanni et al. 2012, Cucchiara et al. 2012, Spinella 2013, Foster et
al. 2013).
An analytical model for evaluation of the N-M-V interaction domain of RC and PC elements by
means of a unified simplified approach was proposed in (Recupero et al. 2005). It has been
derived as the generalization of the series of previously mentioned models based on the stress-
fields approach.
This simplified approach assumes compressive stress field in the concrete, and equivalent
uniformly distributed tensile stresses, corresponding to the action of the stirrups. In this model the
inclination angle  of the compressive stresses may be different from 45° degrees; as a matter of
fact it varies as the shear force increases after yielding of web reinforcements.
Since any analysis exploiting the potential of the proposed approach has not yet been
developed, in this paper a study concerning the optimization of steel reinforcement layouts of PC
elements is carried out, and a procedure for choosing the layouts is proposed.
To illustrate the applicability of the proposed procedure, a design example of a bridge PC beam
is shown. Finally, the beam, designed by this procedure, is virtually loaded up to failure by a
nonlinear FEM code VecTor (Wong and Vecchio 2002), a well known and reliable code based on
the Modified Compression Field Theory (MCFT) (Vecchio and Collins 1986). Preliminarily, the
effectiveness of both the design model and the FEM code will be proved against results of
experimental tests collected in literature.
2. Review of the analytical model
The sectional analytical model described in detail in (Recupero et al. 2005) is herein used to
provide the shear-flexure strength of PC beams. It allows to take into account the simultaneous
presence of bending moment and shear force on the beam including the effect of prestressed
tendons.
The load-carrying capacity of a beam is evaluated by using a simplified layered model, in
which the tensile contribution of concrete is neglected, flanges and the part of the web are assumed
to resist to normal stress only, while the remaining central part of web is able to resist both to
normal stress and shear stress also (Fig. 1a). The flanges and the portion of the web at the top and
the bottom of the cross-section, having depth (z1) and (z2) respectively are subjected to uniform
236

Fig. 1 a) Geometrical model of the beam; b) internal and external forces scheme assumed
stress fields in the longitudinal direction. The remaining part of the web, having a depth equal to
z3, is also subjected to a uniform stress field inclined to the longitudinal direction of the beam by a
variable  angle, due to the shear stresses along cracks. Longitudinal reinforcements of the beam
can be distributed along the entire depth of cross-section (flanges and web), while the transversal
reinforcements is considered only on the web. Both reinforcements give uniform stress fields (Fig.
1b).
Summarizing, normal stresses in concrete and steel rebars in top and bottom flange t tf, t bf and
in ending portions of web w1, and w2, contribute to the axial compression, while the normal and
shear stresses w3 and  along the inner part of web represent actions in the inclined concrete strut
and the stress fields due to transversal and skin reinforcements.
The analytical model allows in predicting the beam failure, which can occur either by concrete
crushing or by reinforcement yielding. The ultimate strength and mode of collapse are obtained
considering a linear domain defined by different limit conditions on the axial and shear stresses in
the flanges and outermost web layers, and in the central web layer, which also take into account
the geometrical percentage of reinforcement (tf, bf, w, tw), and the design value of steel yielding
and compression strength (fyd, fcd1, fcd2) (Eurocode 2002). The axial forces due to prestressing
tendons (pi), acting along a direction inclined by an angle to the longitudinal axis of the beam, are
also limited to the design value (fpd).
Furthermore, some geometrical conditions have to be satisfied. They concern the positive value
of web portion depth (z1, z2, z3) and their minimum values. In particular, the central web layer
depth (z3) must be large enough to carry on shear stresses. A numerical integration procedure along
all layers (Recupero et al., 2005) allows calculation of the resultants of the axial and shear stresses,
including the forces acting on the prestressing tendons, thus their sum provide the internal resisting
force of the cross-section in terms of design resistant bending moment (MRd) and design resistant
shear force (VRd). A resume of equations involved is reported in Appendix A.
The best outcome of the mathematical model is given by a linear programming method that
allows one to evaluate both the depth of the web layers and all stresses by maximizing the element
resisting action in the respect of the plastic admissible conditions on material strengths and
geometrical restrains on the layer sizes.
237

3. Finite element model
To validate the efficiency of the analytical model and the proposed design procedure, the
responses of the specimens that will be designed in this research work will be evaluated by
sophisticated nonlinear tools. The NLFEA will be also validated by reproducing the response of
some experimental tests collected in literature.
The NLFEA allows one to obtain information about the behaviour of structural member
considered. They concern stress and strain conditions at different load stages, providing a wide and
accurate analytical reproduction of tests.
To reproduce the response of RC elements subject to different load conditions, several
analytical formulations have been proposed in literature, which adopt theoretical models based on
several constitutive laws and mechanical theories. The theoretical models taken into account in the
NLFEA utilized herein are the MCFT and the Disturbed Stress Field Model (DSFM), which
represent general models for the load-deformation behaviour of two-dimensional cracked RC
subjected to shear and flexure (Vecchio and Collins 1986). The concrete stresses in the principal
directions are summed with rebar stresses which are assumed to act along rebar longitudinal axis
only. The concrete constitutive behaviour both in compression and tension has been originally
derived from wide experimental survey performed by Vecchio and Collins (1982).
The basic assumption of the MFCT is that the principal strain directions coincide with the
principal stress directions. This assumption has recently been removed by Vecchio (2000) which
has introduced the DSFM. The DSFM explicitly incorporates rigid slipping along crack surfaces
into the compatibility relations for the element, allowing for a divergence between inclination
angles of average principal stress and apparent average principal strain in the concrete. In addition,
MCFT and DSFM have been recently extended to the case of fibrous concrete elements (Colajanni
et al. 2008b, Spinella et al. 2012), proving the ability to reproduce the response of structural
members with different mechanical and load conditions.
In the MCFT the compatibility conditions linking the strains in the cracked concrete with the
strains in the rebars are expressed in terms of average strains, where the strains are measured over
base lengths that are greater than the crack spacing. The equilibrium conditions, which link the
concrete stresses and the rebars stresses to the applied loads, are also expressed in terms of average
stresses.
In the same way, the strains used for the stress-strain relationships are average strains, thus they
consider together the combined effects of local strains at cracks, strains between cracks, bond-slip,
and crack slip. The calculated stresses are also average stresses in that they implicitly encompass
the stresses between cracks, stresses at cracks, interface shear on cracks and dowel action. In this
model, the cracked concrete in RC is treated as a different material with empirically defined
constitutive law. This constitutive behaviour can differ from the traditional stress-strain curve of a
standard cylinder, for example.
The equilibrium equations, the compatibility relationships, the rebars stress-strain relationships,
and the stress-strain relationships for the cracked concrete in compression and tension enable the
average stresses, the average strains, and the angle to be determined for each load level up to
failure.
As can be deduced from the above, the MCFT and DSFM are powerful and general tools for
the NLFEA of RC elements, and for these reasons it has been chosen to perform several numerical
analyses.
238

4. Experimental validation of analytical and fem models
4.1 Experimental tests by literature
The analytical model was validated against several experimental data collected from the
literature. Herein, tests on PC beams performed by Tan and Ng (1998) are taken into account. As
shown in Fig. 2, the specimens have a length between 1200 and 3000 mm, the T-shaped cross
section is 300 mm in depth, and they are externally prestressed with straight tendons. The beams
were identified by the codes ST-1, ST-2, ST-2C, ST-2C+, ST-2S, ST-2P, and ST-3. They were
designed with different concrete strengths, internal reinforcement, and span as test parameters.
Beam ST-2 served as a reference beam. Beam ST-2C and ST-2C+ were similar to ST-2 except for
the concrete strength. Beam ST-2S had lesser shear reinforcement in the form of vertical stirrups
while beam ST-2P was subjected to a midspan load; otherwise they were the same as beam ST-2.
Beams ST-1 and ST-3 differed from beam ST-2 only in shear span, which was one-third of the
effective span in each beam. Each beam was provided with a 100 mm wide deviator at midspan.
Except for beam ST-2C+, the internal longitudinal steel reinforcement consisted of two high-yield
deformed steel bars, 16 mm in nominal diameter (designated as T16 bars), at the bottom and four
mild steel bars, 8 mm in diameter, at the top, with average yield strength of 530 MPa and 338 MPa
respectively. In beam ST-2C+, two high-yield deformed steel bars with a diameter of 20 mm and
an average yield strength of 460 MPa were used as the bottom longitudinal reinforcement instead.
Transverse reinforcement consisting of R8 mild steel stirrups was provided throughout the
length of all beams except ST-3 where mild steel stirrups with a diameter of 6 mm and average
Table 1 Dimensions and reinforcement details of beams tested by Tan and Ng (1998)
Specimen B1 B2
ST-1 0.852 0.883
ST-2 0.898 0.929
ST-2C 0.961 0.994
ST-2C+ 0.935 1.005
ST-2S 0.734 0.755
ST-2P 0.878 0.909
ST-3 0.957 0.986
Mean 0.888 0.923
Standard deviation 0.079 0.087
Table 2 Numerical versus experimental ratios for beams tested by Tan and Ng (1998)
Specimen B1 B2
ST-1 0.852 0.883
ST-2 0.898 0.929
ST-2C 0.961 0.994
ST-2C+ 0.935 1.005
ST-2S 0.734 0.755
ST-2P 0.878 0.909
ST-3 0.957 0.986
Mean 0.888 0.923
Standard deviation 0.079 0.087
239

Fig. 2 Geometrical setup of specimens tested by Tan and Ng (1998)
yield strength of 300 MPa were used. The spacing of the stirrups was 200 mm within the constant
moment region. Within the shear spans, the spacing was 50 mm for beam ST-3, 200 mm for beam
ST-2S and 75 mm for other beams (Tab. 1).
The comparison of the model prediction against test results is carried out neglecting the safety
coefficient proposed by Eurocode (2002) for uniaxial concrete strength fcd1 and for stresses in
presence of shear fcd2. For the latter the effectiveness shear factor is assumed equal to 0.70 and the
partial safety factor for concrete is set equal to 1 [i.e. fcd1 = (1 – fck/250)fck; fcd2 = 0.7 (1 –
fck/250)fck].
With the aim to compare the performances achieved by the analytical model with different
assumptions, two different web depths are considered: in the first case it is equal to net web depth
(B1); in the second one it is extended to the position of the bottom reinforcement in the flange
(B2). These choices reflect the fact that the shear resisting portion of web is not rigorously defined.
In Table 2, the ratios between analytical and experimental results are reported. They show that the
analytical model provides conservative prediction of the load-carrying capacity of specimens, with
the best performance achieved adopting the depth web extended up to the bottom longitudinal
reinforcement (B2). Thus, it confirms that the physical procedure is adequate to reproduce the
experimental results.
4.2 Fe modeling of specimens and load conditions
The specimens of survey tests carried out by Tan and Ng (1998), as described in the previous
section, are modelled developing a two-dimensional plane stress model, suitably restrained to the
symmetrical axis (Fig. 3). The mesh is composed of four-node rectangular elements 50 × 25, 50 ×
35, and 50 × 50 mm in size with uniform thickness to represent the concrete (each region of the
beam is identified by a different colour), and two-nodes truss bars with uniform cross-sectional
area for longitudinal reinforcement. To take into account the action of prestressing cable, an
equivalent nodal force is evaluated by simple geometric considerations and applied at the beam
(Fig. 3). The finite element size is chosen adopting the cover as vertical size and a ratio between
sides of rectangular element close to one. Sliding between rebar and concrete is neglected, and
then perfect steel-to-concrete bond is therefore assumed (Spinella et al. 2012).
According to experimental procedure followed by Tan and Ng (1998), support and point loads
were modelled as concentrate nodal actions on steel plates with a thickness of 25 mm, fully
connected to concrete.
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Fig. 3 Two-dimensional mesh model for specimen ST-2
Numerical analyses are performed assigning three different load patterns acting respectively, in
sequence: a static self weight on all concrete elements; a static nodal force equivalent to the effect
of prestressing cable, applied to the free side of the beam; and a monotonically increasing
displacement, co-axial to the effective transversal load, on the node located in the middle of the
steel transfer load plate. The total load is computed as twice the reaction force at the support. Thus,
the displacement controlled procedure is able to reproduce the post-peak branch of the load-
displacement curve.
The software that implements the MCFT allows managing many parameters concerning
mechanical characteristics of materials and their constitutive laws. The numerical analyses are
carried out assuming the default values for each parameter, and specifically taking the Hognestad
parabola for concrete, and the elastic-plastic law for steel rebars.
4.3 Comparson of results by fem analysis
Figs. 4-6 illustrate the load-displacement numerical curves for the considered beams. The
experimental curves are represented by smooth solid lines, while in each curve provided by the
NLFEA, a different symbol is added. Furthermore, in the same graph, the load capacity (Panl) for
each specimens obtained by analytical model is plotted by a straight dashed line and reported in a
small table also.
The numerical curves show the ability of MCFT to capture the stiffness of those specimens and
to estimate load capacity and ultimate deflection (Tab. 3). Moreover, the slope of some numerical
curves is slightly more inclined than the slope of experimental curves, mainly at the beginning of
the non-linear branch, probably due to the effect of tension stiffening modelling, which plays an
important role at the onset of cracking.
Finally, it can be underlined as the estimate of ultimate capacity, formulated by analytical
model as based on plastic approach, is always on the safety side.
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Table 3 FEM and experimental results for beams tested by Tan and Ng (1998)
Specimen PFEM Pexp PFEM/Pexp
ST-1 373.6 385.0 0.97
ST-2 307.2 310.2 0.99
ST-2C 301.8 299.6 1.01
ST-2C+ 303.0 270.5 1.12
ST-2S 309.8 300.0 1.03
ST-2P 356.4 331.1 1.08
ST-3 186.8 186.8 1.00
Mean / / 1.03
Standard deviation / / 0.05
Fig. 4 Analytical versus experimental load-displacement curves for ST-2, ST-2C, and ST-2C+ beams
Fig. 5 Analytical versus experimental load-displacement curves for ST-2P and ST-2S beams
Fig. 6 Analytical versus experimental load-displacement curves for ST-1 and ST-3 beams
242

5. Proposed design procedure
The proposed procedure to design PC structural elements is powerful and flexible. It allows one
to evaluate the interaction between the principal parameters that govern the physical phenomena.
In fact, after defined the prestressing reinforcement ratio and the longitudinal and transversal web
reinforcement on the basis of the criteria that will be explained afterwards, the interaction between
bending moment, flange reinforcement and shear can be completely evaluated.
In Fig. 7, a design chart for the design of total longitudinal flange reinforcement mechanical
ratio ft, derived from the normalized interaction domain for the beam ST1 is plotted. It shows the
curves of the longitudinal flange reinforcement mechanical ratio ft needed to obtain a given value
of the normalized design bending moment msd; the curves prove that that flange reinforcement
does not contribute to the strength of beam for normalized bending moment close to 1/10. By
varying the bending moment, the shear strength increases with the increase of the flange
reinforcement. Thus, this design chart is a strong and useful tool both in the preliminary design, to
understand the magnitudes of force involved, and in the final check, to optimize the longitudinal
reinforcement ratio of the structural element in order to contribute to the shear strength also.
In first step, the cross-section shape, dimensions and the layout of the prestressing
reinforcements should be designed by governing or limiting deflection, cracking and compression
stresses for service limit state (SLS) stages, as appropriate, in accordance with the code
requirements. The cracking of a beam under service conditions should be controlled, with limits on
crack width being selected to ensure acceptable appearance and durability.
In second step, once the size of the prestressing reinforcement ratio has been settled, the design
of ordinary additional reinforcement, in longitudinal and transversal direction are performed by
control of ultimate limit states (ULS). The longitudinal reinforcement can be placed both in the
flanges and in the web (as skin reinforcement), the transversal reinforcement in form of stirrups is
placed in the web. Generally, several configurations of longitudinal (l) and transversal (w)
reinforcements are admissible in the web. In fact, when longitudinal web reinforcement (l)
increases the transversal reinforcement decreases (w) and vice versa. Therefore, the designer can
choose, among the different configurations that are allowable, the more convenient for practical
and economical requirements. These concepts are graphically explained in the design charts
depicted in Figs. 8-9. They are useful tools that allows the selection of more proper configuration,
among the different ones, by providing the optimal reinforcement web layout for several values of
total longitudinal flange reinforcement (ft).
Fig. 7 Design chart (msd-ft) for the ST-1 beam
243

Fig. 8 Interaction domain (l-w) for section S01 of full-scale bridge beam
Fig. 9 Interaction domain (l-w) for section S02 of full-scale bridge beam
5.1 Example: dsign of a full scale bridge beam
The design procedure described in the previous sections is herein used to perform the design of
a full scale bridge beam, showing the ability of proposed procedure to optimize the amount of skin
reinforcement and stirrups along the web to resist at shear and bending moment. As better
described forward, a new value of mechanical percentage of stirrups has been assumed for each
regions of PC beam, then a value of mechanical percentage of skin reinforcement has evaluated for
a given value of mechanical percentage of flanges reinforcement.
Fig. 10 shows the actual views of bridge taken into account. It consists of seven double “T” PC
beam with single span L = 34.60 m and a RC deck with thick of 200 m. The final load was
estimated considering the following design values: deck weight (93 kN/m); PC beams weight
(139.5 kN/m); transversal RC diaphragms weight (10.5 kN/m); guard rails (6 kN/m); handrails
(2.25 kN/m); and flooring (40.5 kN/m). The sum of static loads was divided for the number of PC
beams to obtain qs = 291.8/7 = 41.7 kN/m. The traffic loads provide a design dynamic equivalent
load (qd) on the lateral PC beam equal to 22.9 kN/m. Finally, the total design load on a single PC
beam is qtot = qs + qd = 40.5 + 22.9 = 64.6 kN/m. In Fig. 11, the transversal section of bridge is
represented, and in Fig. 12 the prestressing reinforcement setup of PC beam is plotted.
The PC beam was previously designed following two phases: firstly, calculating the
prestressing force of cables needed to avoid cracks at the SLS; then, estimating the longitudinal
reinforcement amount of flanges and the transversal reinforcement amount to resist at the shear
force and the bending moment at the ULS. Finally, the minimum skin reinforcement gave by the
244

code was spread along the web of the beam.
The schematic layout of PC beam and prestressing reinforcement forces of half beam is shown
in Fig. 13 (considering the symmetry of the structure) where the prestressing normal force (in kN)
due to strands is represented by arrows in correspondence of the cross-section where it is
completely developed. The geometrical percentage reinforcement is summarized in Table 4, where
sx and sy are the geometric percentage of reinforcement, in x (longitudinal) and in y direction
(transversal), for original (a) and optimized (b) beam setup, respectively; z and t are the depth and
the thickness in mm of the cross-section along the longitudinal axis of the beam, respectively;
finally, Ri (i = 1, 2, 3, 4) is the generic region of length l (mm) in which the PC beam is ideally
sliced.
At the aim to optimize the reinforcement of PC beam, the design procedure described in the
previous sections is used considering some cross-sections along the longitudinal axis of the PC
beam and the related mechanical and geometrical characteristics, as well as the different stress
states. The control cross-sections of beam are chosen with narrow pitch in the vicinity of the
support, where the shear force is relevant, and with wider pitch in the central part of the beam.
The beam is pinned at the support, the total distributed load (qtot = 64.6 kN/m) is smeared along
the deck as an equivalent volume weight; finally, an external vertical displacement applied at one
quarter of the span length (Kani 1967) will be monotonically incremented until the PC beam crisis.
This load scheme allows to simply calculating all forces in each cross-sections of the beam; thus
the sectional method can be used to evaluate the design amounts of stirrups and longitudinal web
reinforcement assuming a constant amount of longitudinal reinforcement in the flanges.
The Figs. 8 and 9 show the design charts relating the mechanical percentage of longitudinal
web reinforcement (l) and the mechanical percentage of transversal reinforcement (w), for
several values of mechanical percentage of flanges reinforcement (ft), evaluated in
correspondence of section S01 (close to the support) and S02 (two meters far from the support),
respectively (Fig. 13). The curves show the interaction between longitudinal web reinforcement
and stirrups, which tends to flatten out with the increase of ft and the shift of the cross-section far
from the support. The former is due to the rule of longitudinal flange reinforcement to resist at
normal stress (normal force and bending moment); the latter is due to the action of cables which
provides an internal bending moment that tends to balance the external bending moment.
(a) (b)
Fig. 10 (a) Longitudinal and (b) transversal view of bridge
245

Fig. 11 Transversal section scheme of bridge (dimensions in cm)
Fig. 12 Prestressing reinforcement setup of PC beam
Table 4 a) Original; and b) optimized geometrical reinforcement of full-scale bridge beam
R1
l = 3000 mm
R2
l = 5000 mm
R3
l = 3500 mm
R4
l = 5800 mm
sx (%) sy (%) sx (%) sy (%) sx (%) sy (%) sx (%) sy (%)
S
z = 200 mm
t = 1800 mm
0.096 0.040 0.096 0.040 0.096 0.040 0.096 0.040
Top
z = 200 mm
t = 500 mm
0.905 0.113 0.905 0.113 0.101 0.057 0.101 0.057
Web
z = 1300 mm
t = 180 mm
a) 0.172
b) 0.395
a) 1.257
b) 0.873
a) 0.172
b) 0.079
a) 0.873
b) 0.873
a) 0.172
b) 0.079
a) 0.436
b) 0.873
a) 0.172
b) 0.079
a) 0.279
b) 0.436
Bottom
z = 300 mm
t = 700 mm
0.096 0.081 0.096 0.081 0.096 0.040 0.096 0.040
Assuming a value of mechanical percentage of stirrups, the correspondent value of mechanical
percentage of skin reinforcement is easily evaluated for a given value of mechanical percentage of
flanges reinforcement (fl). In the case taken into account, at w = 0.22 and for fl = 0.02, the
minimum value of l = 0.10 is obtained for the cross-section close to the support. This minimum
value of mechanical percentage skin reinforcement corresponds at ten longitudinal rebars in 12
mm of diameter along the two sides of the web, and considering the mechanical properties of
concrete (fcd2 = 14.8 MPa) and steel (fyd = 373.9 MPa) the geometrical percentage of skin
246

reinforcement is sx = 0.395%, while for transversal reinforcement two stirrups with two legs in 10
mm of diameter is used. In the same way it can proceed to the design of other cross-sections can
be performed, obtaining the optimized setup scheme of web reinforcement (b) summarized in
Table 4. The optimized scheme shows as the reinforcement is homogeneous along the entire web
and in both the orthogonal directions.
As for the specimens of Tan and Ng (1998), the considered full scale bridge beam is now
analyzed by the NLFEA. The modelling technique used is the same described in the previous
sections, and in Fig. 14a the adopted mesh is shown and indicating with a different colour each
region of the beam. The two-dimensional plane stress model is suitably restrained to the
symmetrical axis and the mesh is composed of four-node rectangular elements 200 × 260 mm in
size with uniform thickness to represent the concrete, and two-nodes truss bars with uniform cross-
sectional area for longitudinal reinforcement. Also in this case, the numerical analyses are
performed assigning three different load cases acting in succession. Several static nodal loads
equivalent to the action of prestressing cable are applied to the free side of the beam representing
the normal force due to the cables. The Fig. 15 shows the P-  curves evaluated by the NLFEA of
the beam as originally designed and as herein optimized in the web reinforcement, respectively.
The target load on half beam is P = qtot L/2 = 1124.5 kN. The curve that represents the numerical
analysis of the original scheme beam provides a load capacity of 1267.1 kN that is 13% greater
than target value. The numerical analysis of the optimized scheme beam provides a load capacity
of 1147.8 kN that is 2% larger than target, thus, a value closer to the target is obtained. This layout
is more advantageous and cheaper than the original one, without any degradation of performance,
as showed by the crack pattern obtained by FEM analysis (Fig. 14b).
Fig. 13 Schematic layout of PC beam and prestressing reinforcement forces
(a)
(b)
Fig. 14 (a) Two-dimensional mesh model and (b) crack pattern for full-scale bridge beam
247

Fig. 15 Analytical load-displacement curve for (a) original and (b) optimized full-scale bridge beam
6. Conclusions
Many models are available in predicting the shear strength of PC beams in literature, and
historically, the large majority of technical papers and codes propose two different models for
reinforced concrete structures on one side and for prestressed elements from the other.
In this paper a unified model, already proposed by the same authors, was scrutinized and
discussed. The model is effective both for RC and for PC elements and, differently from those
existing in literature, is able of taking in account the effects of axial force, bending moment, and
shear force interaction.
In this work, the characteristics of the model were analyzed and its effectiveness is proved by
favourable comparison against the results of experimental tests collected in literature. The results
showed that the proposed model provide a straightforward safe assessment of the ultimate load
with and is a handy tools for design and verification of element strength.
Finally, a procedure for reinforcement design optimization of PC beam was proposed and
applied to a case of study. A full-scale PC beam of an existing bridge was redesigned, obtaining a
more advantageous and cheaper configuration than original one. The two specimens are virtually
conducted up to collapse by a NLFEA. showing the effectiveness of the proposed procedure.
Additional efforts are needed in order to analyze beams with different cross-section shapes and
reinforcement setup to confirm the generality of the proposed model.
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250

Symbols
jf longitudinal flange reinforcement (j = t,b with t = top and b = bottom)
Al longitudinal web reinforcement
Aw transversal web reinforcement
bw web width
fcd1 = 0.85(1-fck/250)(fck/c)= design strength of concrete, for long period uniaxial load (fck in MPa)
fcd2 = 0.60(1-fck/250)(fck/c)= design strength of concrete, in presence of transversal load (fck in MPa)
fck characteristic strength of concrete
fyd design yield strength of ordinary steel reinforcement
fpd design strength of steel prestressing tendons
l length of region
MRd resisting bending moment
MSd design bending moment
P load capacity of specimen
Ri region of beam (i = 1, 2, 3)
S area of whole cross-section
Sjf area of flange (j = t,b with t = top and b = bottom)
Swi area of web layers (i = 1, 2, 3)
s spacing of transversal web reinforcement
t thickness of region
VRd resisting shear force
VSd design shear force
zi depth of web layers (i = 1, 2, 3)
i angle of the ith tendon on the longitudinal direction
 midspan displacement of specimen
c partial safety factor for concrete
s partial safety factor for steel
 angle of compressive stress field on the longitudinal direction
jf = Ajf/Sjf longitudinal flange reinforcement ratio (j = t,b with t = top and b = bottom)
sx,sy longitudinal and transversal reinforcement ratio
wl = Awl/bwhw longitudinal web reinforcement ratio
wt = Awt/bws transversal web reinforcement ratio
jf axial stress in flange layers (j = t,b with t = top and b = bottom)
pi axial stress in the ith tendon
wi axial stress in web layers (i = 1, 2, 3)
 shear stress in the central web layer
fl = (Atf+Abf/S)(fyd/fcd1) total longitudinal flange reinforcement mechanical ratio
l = (Al/bwhw)(fyd/fcd1) longitudinal web reinforcement mechanical ratio
w = (Al/bws)(fyd/fcd2) transversal web reinforcement mechanical ratio
251

A. APPENDIX
The failure of the structural element may occur either by concrete crushing or by reinforcements
yielding. Using the design values given by Eurocode (2002) for steel yielding fyd and for concrete
compression strength fcd1 and fcd2, the following conditions are obtained:
In the top and bottom flange layers
 1cd tf yd tf tf ydf f f      (A.1a)
 1cd bf yd bf bf ydf f f      (A.1b)
In the outermost web layers
 1 1cd l yd w l ydf f f      (A.2a)
 1 2cd l yd w l ydf f f      (A.2b)
In the central web layer
   3 3tan tanw l yd w l ydf f            (A.3a)
cotw ydf   (A.3b)
2 sin coscdf   (A.3c)
Furthermore, each prestressing tendon is subjected to an axial force, acting with an angle of idegrees to
the longitudinal direction, which is taken into account separately from the aforementioned stress fields. The
stress in the ith tendon is limited by the condition:
pi pdf  (A.4)
The resultants of axial and shear stress, plus the effect of the forces acting on the prestressing tendons,
provide the resisting internal actions of the cross section, bending moment MRd and shear force VRd (while
axial force NRd is null):
cos
p
Rd pi pi pi i
nS
M ydS A y     (A.5a)
3
sin
pw
Rd pi pi pi i
nS
V dS A y     (A.5b)
cos 0
p
Rd pi pi pi i
nS
N dS A y     (A.5c)
In the previous equations, the terms related to the areas Sw1, Sw2, and Sw3 depend on the depth of the web
layers z1, z2, and z3, which may vary according to the following geometrical and static conditions:
1 0z  (A.6a)
2 0z  (A.6b)
1 2 3 wz z z h   (A.6c)
3 3minz z
(A.6d)
In particular, Eq. (A.6d) states that the central web layer depth must be sufficient to bear shear stresses;
its minimum value z3min, which depends on concrete strength fcd2 and on transverse web reinforcement
252

mechanical ratio t,, is given by:
 
3min
2 1
Sd
cd w t t
V
z
f b  


if 0.5t  (A.7a)
3min
2
2 Sd
cd w
V
z
f b
 if 0.5t  (A.7b)
The web depth hw has to be greater than z3min to satisfy the shear stress equilibrium. In case z3min > hw the
web of the beam is not sufficient to sustain the shear stress and part of flange can be assumed to sustain
shear stress in place of longitudinal stress, but in any case z3min cannot be greater than the full depth of cross-
section without enlarging the cross-section width.
253
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