Flanged sections are often used for longspan concrete beams to maximize their structural efficiency. However, although for the same sectional area a flanged section could render a higher flexural strength, it would also lead to a lower flexural ductility, especially when heavily reinforced. Thus, when evaluating the flexural performance of a beam section, both the flexural strength and ductility need to be considered. In this study, the postpeak flexural behaviour of flanged sections is evaluated by means of an analytical method that uses the actual stress–strain curves of the materials and takes into account strain reversal of the tension reinforcement. From the numerical results, the flexural strength–ductility performance of flanged sections is investigated by plotting the strength and ductility that could be simultaneously achieved in the form of design graphs. It is found that (1) at the same overall dimensions and with the same amount of reinforcement provided, a flanged section has lower flexural ductility than a rectangular section; (2) at the same overall dimensions, a flanged section has inferior strength– ductility performance compared to a rectangular section; and (3) at the same sectional area, a flanged section has better strength–ductility performance compared to a rectangular section. Copyright © 2004 John Wiley Sons, Ltd.
Trang 1FLEXURAL STRENGTH–DUCTILITY PERFORMANCE
OF FLANGED BEAM SECTIONS CAST OF
HIGH-STRENGTH CONCRETE
A K H KWAN* AND F T K AU
Department of Civil Engineering, University of Hong Kong, Hong Kong
SUMMARY Flanged sections are often used for long-span concrete beams to maximize their structural efficiency However, although for the same sectional area a flanged section could render a higher flexural strength, it would also lead
to a lower flexural ductility, especially when heavily reinforced Thus, when evaluating the flexural performance
of a beam section, both the flexural strength and ductility need to be considered In this study, the post-peak flex-ural behaviour of flanged sections is evaluated by means of an analytical method that uses the actual stress–strain curves of the materials and takes into account strain reversal of the tension reinforcement From the numerical results, the flexural strength–ductility performance of flanged sections is investigated by plotting the strength and ductility that could be simultaneously achieved in the form of design graphs It is found that (1) at the same overall dimensions and with the same amount of reinforcement provided, a flanged section has lower flexural ductility than a rectangular section; (2) at the same overall dimensions, a flanged section has inferior strength– ductility performance compared to a rectangular section; and (3) at the same sectional area, a flanged section has better strength–ductility performance compared to a rectangular section Copyright © 2004 John Wiley & Sons, Ltd.
1 INTRODUCTION When a reinforced concrete beam section is subjected to flexure, the applied bending moment is resis-ted jointly by the compressive force developed in the concrete and the tensile force developed in the steel reinforcement, which are equal and opposite to each other and together form a couple, as illus-trated in Figure 1(a) Let the compressive force in the concrete and the tensile force in the
reinforce-ment be denoted by C and T respectively and the lever arm, i.e the distance between C and T, be
denoted by a Since the applied bending moment is always equal to C or T times a, for given C and
T, the flexural strength of the beam section increases with the lever arm a Hence, a larger lever arm would lead to a better structural efficiency of the beam section One way to increase the lever arm is
to increase the depth of the beam section, but very often the depth of a beam section is limited because
of the headroom or other geometric requirements and cannot be increased without affecting the overall layout of the structure If the depth of the beam section cannot be increased, an alternative is to change the beam section to a flanged section, as shown in Figure 1(b) In a flanged section, such as a T- or box-shaped section, most of the concrete is located near the extreme compression fibre and thus the
line of action of C is at a larger distance from that of T Because of the larger lever arm in a flanged
section than in a rectangular section, the structural efficiency of a flanged section is generally larger than that of a rectangular section
Published online 9 June 2004 in Wiley Interscience (www.interscience.wiley.com) DOI:10.1002/tal.231
Copyright © 2004 John Wiley & Sons, Ltd Received December 2002
* Correspondence to: Professor A K H Kwan, Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong E-mail: khkwan@hku.hk
Trang 2It is sometimes argued that the concrete near the geometric centre of the section is of little use because it develops small compressive stresses and contributes little to the flexural strength, and that the higher structural efficiency of a flanged section is attained mainly by relocating the concrete near the centre of the section to the flange area near the extreme compression fibre where much higher compressive stresses would be developed (Ferguson, 1981) In actual fact, apart from contributing to the shear and torsional strengths of the beam section, the concrete near the centre also contributes sig-nificantly to the flexural ductility of the beam section At the post-peak stage, when the peak bending moment has already been reached and the moment resisting capacity of the beam section is decreas-ing, the concrete near the extreme compression fibre would gradually lose its strength and the neutral axis of the section would move towards the tension reinforcement Consequently, the concrete near the centre would at the post-peak stage develop much higher compressive stresses than before and contribute significantly to the residual moment resisting capacity of the section Because of this, a rec-tangular section, which has relatively more concrete near the centre, is generally more ductile than a
T
C
l a
distribution
Stress distribution (a) A rectangular section subjected to flexure
T
C
distribution
Stress distribution
l a
(b) A flanged section subjected to flexure Figure 1 Beam sections subjected to flexure
Trang 3flanged section In other words, although a flanged section has a higher structural efficiency in terms
of flexural strength, it might at the same time have a somewhat lower flexural ductility
Since ductility may also be a governing factor determining the safety of a structure, when evaluat-ing the structural performance of a beam both the flexural strength and ductility need to be consid-ered Maximizing the flexural strength or the structural efficiency in terms of flexural strength without paying proper attention to the flexural ductility may not produce the best overall structural perform-ance The best way to evaluate the flexural strength–ductility performance of a beam section is to eval-uate the flexural strength and flexural ductility that could be simultaneously achieved at various levels
of reinforcement, as will be illustrated in this paper However, while the flexural strength of a beam section can be determined quite easily using the ordinary beam bending theory, it is not possible to evaluate the flexural ductility using any simple analytical method To evaluate the flexural ductility
of a beam section, it is necessary to carry out non-linear flexural analysis, extended well into the post-peak range so that the ability of the beam section in withstanding inelastic curvature without excessive loss in flexural strength may be determined Such kind of analysis is highly non-linear and involves stress path dependence of the constitutive materials during strain reversal at the post-peak stage
In most books on reinforced concrete design, the analysis of flanged beams is just treated as a straightforward extension of that for rectangular beams That is probably the reason why, compared with investigations on the structural behaviour of rectangular beams, much less work has been done
on flanged beams Moreover, the previous work on flanged beams generally concentrated on aspects
other than the post-peak flexural behaviour For example, Swamy et al (1973) studied the shear
resist-ance of T-beams with varying flange widths and found that the shear resistresist-ance of T-beams was significantly increased by the flange width, the percentage of tension steel and the amount of web
reinforcement Desayi et al (1978) conducted tests of reinforced concrete T-beams and rectangular
beams to study the influence of the flange on the torsional strength of reinforced concrete T-beams The results indicated that the torsional strength contribution of the flange may be estimated by the plastic theory of torsion Prakash Rao (1982) presented a comprehensive summary of the research done in Europe on the design of webs and web–flange junctions under combined bending and shear,
taking into account the interaction between various forces acting on the section Subedi et al (1992)
and Subedi (1993) carried out experimental work focusing on the failure behaviour of thin-walled reinforced concrete flanged beams and observed that the major modes of failure were flexure, diago-nal splitting and web crushing So far, there has been little research on the post-peak flexural behav-iour of flanged beams
At the University of Hong Kong, a new method of non-linear flexural analysis that uses the actual stress–strain curves of the materials and takes into account stress path dependence of the tension reinforcement has recently been developed and applied to study the post-peak flexural behaviour of
singly and doubly reinforced rectangular beam sections (Ho et al., 2003; Pam et al., 2001) It has been found from these studies that at the post-peak stage both the line of action of C and the neutral axis
of the section would gradually shift towards the tension reinforcement and then at a certain point the axial strain of the tension reinforcement would start to decrease, causing strain reversal in the tension reinforcement and consequently stress path dependence of the tensile stress developed therein Using this newly developed method, the effects of various structural parameters including the concrete grade, the tension and compression steel yield strengths and the tension and compression steel area ratios on
the flexural ductility of rectangular beam sections have been quite thoroughly studied (Kwan et al.,
2003)
In the present study, the aforementioned method of analysis has been extended to deal with flanged sections Using the extended analysis method, the non-linear flexural behaviours of typical flanged
Trang 4sections have been analysed and compared to those of rectangular sections with the same overall dimensions After gaining a better understanding of the non-linear flexural behaviour of flanged sec-tions, the flexural strength and ductility that could be simultaneously achieved by flanged sections of different shapes have been evaluated From the numerical results obtained, the flexural strength–duc-tility performance of flanged sections has been studied by plotting the flexural ducstrength–duc-tility against the flexural strength in the form of graphs that may in fact be used as design charts The results revealed that in the design of flanged beam sections more attention to the provision of sufficient flexural duc-tility is generally needed
2 METHOD OF ANALYSIS The constitutive model for unconfined concrete developed by Attard and Setunge (1996), which has been shown to be applicable to a broad range of concrete strength from 20 to 130 MPa, is adopted in the moment–curvature analysis The stress–strain curve of the constitutive model is given by
(1)
in which scand ec are the compressive stress and strain at any point on the stress–strain curve, f coand
eco are the compressive stress and strain at the peak of the stress–strain curve, and K1and K2are
coef-ficients dependent on the concrete grade It should be noted that f co is actually the in situ compressive
strength, which may be estimated from the cylinder or cube compressive strength using appropriate conversion factors Figure 2(a) shows some typical stress–strain curves so derived
For the steel reinforcement, a linearly elastic–perfectly plastic stress–strain curve is adopted Since there could be strain reversal in the steel reinforcement at the post-peak stage despite monotonic
increase of curvature (Ho et al., 2003; Pam et al., 2001), the stress–strain curve of the steel is
stress-path dependent It is assumed that when strain reversal occurs, the unloading stress-path of the stress–strain curve is linear and has the same slope as the initial elastic portion of the stress–strain curve Figure 2(b) shows the resulting stress–strain curve of the steel reinforcement
Only three other basic assumptions have been made in the analysis: (1) plane sections before bending remain plane after bending, (2) the tensile strength of concrete is negligible, and (3) there is
no bond-slip between concrete and steel These assumptions are widely accepted in the literature (Park and Paulay, 1975) The moment–curvature behaviour of the beam section is analysed by applying pre-scribed curvatures to the beam section incrementally starting from zero At a prepre-scribed curvature, the strain profile in the section is first evaluated based on the above assumptions From the strain profile
so obtained, the stresses developed in the concrete and the steel reinforcement are determined from their respective stress–strain curves The stresses developed have to satisfy the axial equilibrium con-dition, from which the neutral axis depth is evaluated by iteration Having determined the neutral axis depth, the resisting moment is calculated from the moment equilibrium condition The above proce-dure is repeated until the curvature is large enough for the resisting moment to increase to the peak and then decrease to 50% of the peak moment Details of the analysis procedure have been presented
in Ho et al (2003) and Pam et al (2001).
The method of analysis previously applied to solid rectangular sections is extended to deal with flanged sections by taking the width of the section as a variable instead of a constant For instance, in the following equations governing the axial and moment equilibrium conditions of the flanged beam section shown in Figure 3:
c co
c co c co
c co c co
2
2
Trang 5(2) (3)
where P is the applied axial load (compressive force taken as positive), M is the resisting moment (sagging moment taken as positive) and b is the width of the section at x from the neutral axis, the width b is taken as a variable during the numerical integration Because of the variable width of the
flanged section, a more sophisticated numerical integration technique has to be used when integrating over the concrete section to determine the axial force and the resisting moment contributed by the stresses developed in the concrete In the present study, Romberg integration (Gerald and Wheatley, 1999), which can significantly improve the accuracy of the simple trapezoidal rule when the integrand
is known at equispaced intervals, has been adopted
d
sc sc n st st n
n
d
sc sc st st
n
0
(a) Concrete
y
ss = f y
ss =E s (es -ep )
ep
Strain unloading
(b) Steel reinforcement
0 10 20 30 40 50 60 70 80
Strain
fco=30MPa fco=50MPa fco=70MPa
f co
f co
f co
Figure 2 Stress–strain curves of concrete and steel reinforcement
Trang 63 FLEXURAL BEHAVIOUR OF TYPICAL FLANGED SECTIONS
3.1 Sections analysed
For the sake of comparing the non-linear flexural behaviour of flanged beam sections to that of a solid rectangular beam section with the same overall dimensions, three beam sections have been analysed,
as shown in Figure 4 They have the same overall dimensions of B = 1200 mm and D = 1500 mm and
D
Section
Stress distribution
Neutral axis
d n
sst
ssc
D 1
D f
x B
B w
A sc
A st
b
Figure 3 Analysis of flanged section
(Note: In all sections, A st = 40,000 mm2)
D f
B =1200mm
B w =400mm
B =1200mm
B w =800mm
B =1200mm
Figure 4 Typical flanged sections analysed
Trang 7are named Section A, Section B and Section C Section A is a T-shaped section with web breadth and
flange depth given by B w = 400 mm and D f = 400 mm respectively, while Section B is a T-shaped
section with web breadth and flange depth given by B w = 800 mm and D f= 400 mm respectively On the other hand, Section C is a solid rectangular section In each section, the same amount of tension
reinforcement given by A st= 40,000 mm2
is provided However, no compression reinforcement is
pro-vided in any of the three sections For a preliminary study, the in situ concrete compressive strength
f co is fixed at 50 MPa, while the yield strength f yt and Young’s modulus E sof the steel reinforcement are fixed at 460 MPa and 200 GPa respectively
The complete moment–curvature curves obtained for the three beam sections are shown in Figure 5
It is seen that the three sections, each provided with the same amount of tension reinforcement, have similar peak resisting moments They all fail in tension (i.e the tension reinforcement yields before the concrete fails) and are therefore under-reinforced That is why all the three sections fail in a ductile manner, as evidenced by the presence of a flat yield plateau in each of their moment–curvature curves Nevertheless, it is obvious that the flexural ductility of Section A is lower than that of Section B, while the flexural ductility of Section B is lower than that of Section C Bearing in mind that the three sec-tions have the same overall dimensions and the same amount of reinforcement provided and that they differ from one other only in the web breadth, it is evident that although the concrete in the web con-tributes little to the peak resisting moment, it does contribute to the residual resisting moment at the post-peak stage and thus the flexural ductility of the section
4 FLEXURAL DUCTILITY OF FLANGED SECTIONS
4.1 Flexural ductility evaluation
The flexural ductility of the beam section may be evaluated in terms of a curvature ductility factor m defined by
0
5
10
15
20
25
30
Curvature 0-3 radian/m)
Section B Section A
Section C
10 ( X1 Figure 5 Moment–curvature curves of beam sections analysed
Trang 8(4) where fuand fyare the ultimate curvature and yield curvature respectively The ultimate curvature fu
is taken as the curvature of the beam section when the resisting moment of the beam section has, after
reaching the peak value of M p , dropped to 0·8 M p On the other hand, the yield curvature fyis taken
as the curvature at the hypothetical yield point of an equivalent linearly elastic–perfectly plastic system
with an elastic stiffness equal to the secant stiffness of the section at 0·75 M pand a yield moment
equal to M p
4.2 Flexural ductility of flanged sections
When comparing the flexural ductility of a flanged section to that of a rectangular section with the same overall dimensions, it is necessary to take into account also the other structural parameters such
as the concrete grade and the amount of tension reinforcement provided because the flexural ductil-ity varies significantly with these parameters The flanged and rectangular sections shown in Figure 4
are reanalysed using different values of f co and A st To study the effect of the concrete grade, f cois set
equal to 30, 50 or 70 MPa To study the effect of the amount of tension reinforcement provided, A stis varied from 10,000 to 80,000 mm2 Figure 6 shows the variation of the curvature ductility factor m
with the tension steel area A stfor the beam sections analysed at different concrete grades
It is seen that the flexural ductility of a beam section, regardless of the sectional shape, decreases
as the tension steel area increases In general, at the same concrete grade and the same tension steel area, the flexural ductility of a flanged section is lower than that of a rectangular section with the same overall dimensions The difference in flexural ductility is relatively small when the beam sections are lightly reinforced but could be quite significant when the beam sections are heavily reinforced In fact, when the beam sections are heavily reinforced, even though the sections are provided with the same amount of reinforcement, it could happen that a flanged section is already over-reinforced while a rec-tangular section still remains under-reinforced So when the beam sections are heavily reinforced with
m=f fu y
0
5
10
15
20
25
Tension steel area, A st (mm2)
fco=30MPa Section A fco=30MPa, Section B fco=30MPa, Section C fco=70MPa, Section A fco=70MPa, Section B fco=70MPa, Section C
f co
f co
f co
f co
f co
f co
Figure 6 Variation of curvature ductility factor m with tension steel area Ast
Trang 9the same amount of reinforcement, a change in sectional shape from rectangular to flanged shape could lead to a change of failure mode from ductile tension failure to brittle compression failure This is because in general a flanged section has a smaller balanced steel area (the tension steel area at which balanced failure occurs) than a rectangular section
It is also evident from Figure 6 that at the same tension steel area the flexural ductility increases slightly with the concrete grade, albeit a higher-grade concrete should be less ductile This may also
be explained by looking at the degree of the beam section being under- or over-reinforced The balance steel area is larger when the concrete grade is higher Thus, relatively, at the same tension steel area, the tension to balanced steel ratio (the ratio of the tension steel area to the balanced steel area) is smaller when a higher-grade concrete is used The tension to balanced steel ratio may be interpreted
as a measure of the degree of the beam section being under/over-reinforced When the tension to bal-anced steel ratio is smaller, the degree of the beam section being under-reinforced is higher and the degree of the beam section being over-reinforced is lower Therefore, at the same tension steel area, the degree of the beam section being under-reinforced increases with the concrete grade and as a result the flexural ductility increases slightly with the concrete grade
From the above, it is obvious that one major structural parameter determining the flexural ductility
is the tension to balanced steel ratio, which is denoted hereafter by l Figure 7 shows the variation of the curvature ductility factor m with the tension to balanced steel ratio l for the beam sections analysed
at different concrete grades It is seen that regardless of the concrete grade and the shape of the beam section, m decreases as l increases and then remains roughly constant when l> 1·0 In general, at the same concrete grade and the same tension to balanced steel ratio, the m-value of a flanged section is slightly higher than that of a rectangular section with the same overall dimensions when l < 0·85 (when the sections are lightly reinforced) and the m-value of a flanged section is slightly lower than that of a rectangular section with the same overall dimensions when l> 0·85 (when the sections are heavily reinforced) Nevertheless, since a flanged section has a smaller balanced steel area compared
to that of a rectangular section, at the same tension steel area, a flanged section has a higher tension
to balanced steel ratio and therefore a lower flexural ductility, especially when the section is heavily reinforced Figure 7 also reveals that at the same tension to balanced steel ratio, regardless of the
sec-0 5 10
15
20
25
Tension to balanced steel ratio l
fco=30MPa Section A fco=30MPa, Section B fco=30MPa, Section C fco=70MPa, Section A fco=70MPa, Section B fco=70MPa, Section C
f co
f co
f co
f co
f co
f co
Figure 7 Variation of curvature ductility factor m with tension to balanced steel ratio l
Trang 10tional shape, the flexural ductility decreases as the concrete grade increases, for the simple reason that
a higher-grade concrete is generally less ductile
5 FLEXURAL STRENGTH–DUCTILITY PERFORMANCE OF FLANGED SECTIONS
In order to study the effect of sectional shape on the flexural strength–ductility performance, a number
of beam sections with different B w /B and D f /D ratios have been analysed using the method developed herein All together, 16 flanged sections with B w /B and D f /D ratios ranging from 0·1 to 0·4 and one rectangular section with B w /B = 1 and D f /D= 1 have been analysed All the beam sections analysed
have the same overall dimensions of B = 1200 mm and D = 1500 mm For each beam section, the con-crete strength f cois set equal to 30, 50 or 70 MPa and the tension to balanced steel ratio is varied from
0·3 to 1·2 However, to reduce the number of variables, the tension steel yield strength f yt is fixed at
460 MPa
5.2 Flexural strength–ductility performance at same overall dimensions
For a given beam section, regardless of whether it is a rectangular section or a flanged section, the use of a higher-tension steel area leads to a higher flexural strength but a lower flexural ductility whereas the use of a lower-tension steel area leads to a higher flexural ductility but a lower flexural strength Hence, the increase in flexural strength obtained by using a higher-tension steel area is achieved at the expense of a lower flexural ductility and the increase in flexural ductility obtained by using a lower tension steel area is achieved at the expense of a lower flexural strength The achieve-ment of both high flexural strength and high flexural ductility is not easy If a beam section can attain high flexural strength and high flexural ductility at the same time, it is said to have a good flexural strength–ductility performance
The flexural strength–ductility performance of a beam section is best revealed by plotting the flex-ural strength and the flexflex-ural ductility that could be simultaneously achieved in the form of graphs Figure 8 shows the flexural strength–ductility graphs so obtained for the beam sections analysed In
the figure, the x-ordinate is the flexural strength, expressed in terms of M p /BD2
, while the y-ordinate
is the curvature ductility factor m The purpose of expressing the flexural strength in terms of Mp /BD2
is to render it independent of the actual dimensions and allow comparison of the flexural strength– ductility performances of beam sections having the same overall dimensions but different shapes From the figure, it can be seen by comparing the curves for flanged sections to those of rectangu-lar sections that in general the flexural strength–ductility curve of a flanged section is lower than that
of a rectangular section Hence, at the same overall dimensions, the flexural strength–ductility per-formance of a flanged section is inferior to that of a rectangular section In other words, although at the same overall dimensions, a flanged section has a smaller sectional area and is thus lighter than a rectangular section, its flexural strength–ductility performance is not as good as that of a rectangular
section Comparing the curves for flanged sections with different B w /B and D f /D ratios, it is evident that when the sections are lightly reinforced the ratio D f /D has greater influence on the flexural strength–ductility performance, while the B w /B ratio has relatively little influence, but when the sec-tions are heavily reinforced both the B w /B and D f /D ratios have significant influences on the flexural
strength–ductility performance
It can also be seen by comparing the curves for beam sections cast of concrete of different grades that for the same sectional shape and dimensions those beam sections cast of a higher-grade concrete