Shear Mechanism of Reinforced Concrete T-Beams with Stirrups

Journal of Advanced Concrete Technology Vol. 5, No. 3, 395-408, October 2007 / Copyright © 2007 Japan Concrete Institute 395

Scientific paper

Shear Mechanism of Reinforced Concrete T-Beams with Stirrups Withit Pansuk1 and Yasuhiko Sato2

Received 11 April 2007, accepted 30 June 2007

Abstract As is well known, in the current design code, the shear strength of beams can be calculated based on the modified truss theory, which cannot take into account the effects of the top flange area of T-beams. Reported experimental data show that the top flange has an effect on the shear capacity of T-beams with shear reinforcement. To predict the shear capacity of T-beams more precisely, the effect of the concrete top flange area on the shear resisting mechanism must be clarified. Comparison of test results for rectangular and T-beams yielded insights into the shear resisting mechanism of T-beams. Verification and clarification of the shear resisting mechanism of T-beams were performed based on the 3D nonlinear finite element code (CAMUI). Finally, a simplified method for determining the failure criteria for shear of RC T-beams is proposed.

1. Introduction

In the slab-beam-girder construction system, the beams are usually built monolithically with a slab. Hence, the portion of concrete slab effectively connected together with a beam can be considered as the flange projecting from each side of the beam. At the same time, the part of the beam at the bottom of the slab is working as the web or stem of the T-shaped beam (hereafter, “T-beams”). This type of beam is widely used in engineering struc-tures, for example RC bridges and building structures.

A monolithic multiple T-section, which has several stems and includes the entire one-way slab that spans transversely between the stems as its flange, is usually used in the construction system. For design work, the multiple T-section is divided into individual T-sections that have a portion of the slab as flanges. For negative bending moment, the flange is on the tension side of the neutral axis, and thus the T-section is in effect a rectan-gular section. For positive bending moment, the flange does provide considerably more compression area. Just how much of the slab projecting from the stem may be considered as part of the individual T-section, as well as the methods of design and analysis for flexural strength, are clearly defined and appear in the current design codes (ACI 1999; Eurocode 2 1992; JSCE 2002). On the other hand, for the shear issue, the situation is completely different. As is well known, in the current design codes, shear strength can be calculated based on the modified truss theory, which cannot take into account the effects of the top flange of T-beams. However, the literature in-cludes reports that the area of the top flange has an effect on the shear capacity and resisting mechanism of

T-beams (Taub and Neville 1960; Bresler and Macgregor 1967; Moayer and Regan 1974). Only a small number of studies have so far been done in an empirical way to consider the existence of top flanges in design (Leon-hardt 1965; Regan and Placas 1970; Hoang 1997). To predict the shear capacity of T-beams more precisely, the effect of top flanges on the shear resisting mechanism must be clarified.

This study first examines how the top flange affects the shear capacity of RC T-beams with and without shear reinforcement based on data analysis using previous experimental results. The comparison of the test results of rectangular and T-beams yielded insights into the shear resisting mechanism of T-beams. Verification and clarification of the shear resisting mechanism of T-beams were performed using a home-made nonlinear finite element program. Finally, a simplified method for de-termining the failure criteria for shear of RC T-beam is proposed.

2. Shear capacity of RC T-beams observed in previous studies

To confirm the factors and variables that affect the shear-resisting mechanism of beams, the data observed in the previous studies were collected and divided into small groups with similar beam parameters, and only the data of T-beams with different flange widths were con-sidered. The selected data included only simple-span beams that failed in shear failure mode under one or two symmetrical concentrated loads. The relevant informa-tion pertaining to sources of data, geometry, materials, and number of beams is summarized in Table 1.

From Fig. 1, in the case of T-beams without shear re-inforcement, the ratio of flange width to web width (bf/bw) has almost no effect on the ultimate shear capacity of T-beams. In the case of T-beams without shear rein-forcement whose a/d ratio is greater than 3.4, it was reported that beams failed immediately after the occur-

1Doctoral course student, Division of Built Environment, Hokkaido University, Japan. E-mail:[email protected] 2Associate Professor, Division of Built Environment, Hokkaido University, Japan.

396 W. Pansuk and Y. Sato / Journal of Advanced Concrete Technology Vol. 5, No. 3, 395-408, 2007

rence of a diagonal shear crack without compression failure in the compression zone. As a result, the flange can be considered not to affect the shear capacity of T-beams. Fig. 2, on the other hand, shows that in the case of T-beams with shear reinforcement, shear strength of

reinforced concrete T-beams increases as the ratio of flange width to web width becomes large. As a result, this study will focus only on the effect of concrete top flanges on the shear resistance of T-beams with shear reinforcement in webs whose a/d ratio is greater than 2.4.

3. Experimental program

3.1 Specimens and materials There were no existing tests for T-beams using the presence of the flange area as the experimental parameter. Also, the comparison of important beam behavior such as concrete strain and stress development in the shear reinforcement between rectangular and T-beams could not be found in previous research. As a result, two rein-forced concrete beams of rectangular and T-shaped sec-tions were tested in this study. The rectangular beam measured 3800 × 150 × 350 mm (length × width × height) and had an effective depth of 300 mm. The cross section of the T-shaped section was almost the same as that of the rectangular beam; only a concrete flange was attached in the top position of a whole long beam. The cross sections of both specimens are shown in Fig. 3. The stirrups in the tested part had a spacing of 110 mm, while stirrups were placed more heavily in the remaining parts of the beam to ensure shear failure within the tested part. The full details of their dimensions, arrangement of re-inforcing steel and loading condition are shown in Fig. 4.

Both specimens had the same tension and compression reinforcement, four D25 and two D10 bars, respectively. Shear reinforcement was D6 stirrup with a closed-hoop shape. The concrete cylinder strength (fc’) for each specimen was 35 MPa. The tension reinforcement ratio (ρl) and shear reinforcement ratio (ρw) were 4% and 0.4%, respectively. The properties of the steel used are given in Table 2.

Table 1 Experimental data of T-beams.

Test series a/d ρwfw (MPa)

Amount of data

Moayer and Regan (1974) 2.4 - 3.5 0.7 - 1.1 4

Ferguson and Thompson (1953) 3.4 - 6.2 - 8

Al-Alusi (1957) 3.4 - 6.5 - 8 Placas and Regan (1971) 3.4 - 3.6 0.4 - 1.2 19

Withey (1908) 3.0 0.5 - 1.4 8 Taub and Neville (1960) 3.0 0.3 - 1.3 7

Note: a = shear span; d = effective depth; ρw = shear reinforcement ratio; fw = yield strength of shear rein-forcement; fc’ = concrete compressive strength = 11 to 57 MPa; ρl = tension reinforcement ratio = 0.01 to 0.05

4 50

1

2

3

4

5

bf/bw

Shea

r stre

ngth

(MPa

) a/d = 3.4-3.6a/d = 5.5-6.5

Fig. 1 Shear strength of T-beams without shear rein-forcement.

2 3 40

1

2

3

4

5

a/d = 3.0-3.6, ρwfw = 0.3-0.7a/d = 2.4-3.6, ρwfw = 1.0-1.4

bf/bw

Shea

r stre

ngth

(MPa

)

Fig. 2 Shear strength of T-beams with shear reinforce-ment.

15075

S2

300

150

S1

Covering = 20

Fig. 3 Cross sections (unit: mm).

Table 2 Properties of bars.

Bar No.

Area (mm2)

Yield point (MPa)

Elastic modulus (MPa)

6 31.67 300 165000 10 71.33 360 192000 25 506.7 400 178000

W. Pansuk and Y. Sato / Journal of Advanced Concrete Technology Vol. 5, No. 3, 395-408, 2007 397

3.2 Test method and measurement Both specimens were tested in a 1000 kN capacity hy-draulic testing machine in a simply supported condition over a span of 3000 mm. The load was applied through a steel loading beam with the spherical bearing unit at both load points. Steel plates 90 mm wide by 15 mm thick were used to distribute loads and support reactions. For both specimens, the load was applied in 10-kN incre-ments until 180 kN and released until zero. The load was applied in the same increments again until failure. After each load increment was stabilized, strains in the tension reinforcement and stirrup, concrete strains in the top part of the beam, and deflections were measured, and crack patterns were noted.

Strain gages were attached to measure strain in each stirrup at distances of 60, 130 and 220 mm above the centroid of the tension reinforcement for both specimens. Also, strain gages were attached to measure strains in tension bars at distances of 350, 525, 700, 965 and 1500 mm from the support for checking yielding of the bars and drawing the strain distribution on the beam sections. The locations of the strain gages and reference number of stirrups for both specimens are shown in Fig. 5. In the experiment, two types of strain gages for concrete, placed inside the concrete and on the concrete surface, were installed into both specimens for measuring strains in the direction parallel to the tension bars. The measured sections are at the area near the loading plate (at the location of stirrup No. 1 in Fig. 5) and the middle of the shear span (at the location of stirrup No. 5 in Fig. 5). The locations of the concrete strain gages at both sections are shown in Fig. 6.

4. Outline of finite element analysis

4.1 Outline of analysis In the present study, a three-dimensional nonlinear finite element code (CAMUI) developed at Hokkaido Univer-sity was used. In this analysis, three-dimensional 20 node iso-parametric solid elements, with 8 Gauss points were adopted for the representation of the reinforced concrete elements. The nonlinear iterative procedure was con-trolled by the modified Newton-Raphson method. In this

procedure, the convergence was satisfied by the ratio of Σ (Residual force)2 to Σ (Internal force)2 becoming less than 10-6. Also, the calculation process was moved to the next step when the ratio was not satisfied over 200 cycles of repetition.

The 3D elasto-plastic and fracture model (Maekawa et

3000 400 50@8

1050 1050 900

350

P/2

450 110@9

P/2

75@14

Fig. 4 Loading condition and rebar arrangement (unit: mm).

Stirrup No.

Strain gage on stirrup

Strain gage on tension bar

1 2 3 4 5 6 78

Fig. 5 Location of strain gages and reference number of stirrups.

X

Y

Z

1525 70

55

50 15

= Gages placed on

surface

= Gages placed

inside concrete

Fig. 6 Locations of gages at sections of stirrup No. 1 and 5 (unit: mm).


al. 1993) that considers the effect of confinement, de-formability and bi-axial compression in the concrete constitutive law was used for the concrete model before cracking. In this model, stresses and strains were repre-sented by an equivalent stress and equivalent strain, respectively. The adopted failure criteria that acted in agreement with Niwa’s model in tension-compression domains and Aoyagi and Yamada’s model in ten-sion-tension domains were extended to 3D criteria by satisfying boundary conditions (Takahashi et al. 2005).

When the first crack occurs, the strains in the global coordinate system are transformed into the strains in the local coordinate system (called crack coordinate system). In the case of the second crack, one of the two axes in the plane coincides with the direction of the intersecting line between the first and second crack plane. Two local systems share one of their axes and another axis in the plane is in the direction where the crack opens. Consti-tutive models were applied in the directions parallel and normal to the crack planes. After calculating stresses from the strains in the crack coordinate system, the stresses were retransformed into stresses in the global coordinate system and superimposed.

The tension-softening model proposed by Reinhardt et al. (1986) is adopted. The model is expressed as the following relationship between σ, the tensile stress car-ried by concrete, and δ, the crack opening displacement.

( ) ( )231

002

3

01 exp1exp1 cccc

ft−+−

−

+=

δδ

δδ

δδσ (1)

where c1 = constant, 3.00 for normal concrete; c2 = con-stant, 6.93 for normal concrete; δ0 = the critical crack opening (crack width at zero stress), 0.16 mm; and ft = concrete tensile strength.

This model is a function of stress-crack opening dis-placement. Consequently, strain in the direction normal to the crack is transformed into crack opening dis-placement by multiplying the crack spacing. The crack spacing observed in the experiment was used in this study.

The ascending part of the Vecchio & Collins model (Collins et al. 1996) was applied for the two-dimensional concrete model in a plane parallel to the crack, as shown in Fig. 7.

−

=

2

00

'2

εε

εε

βσ cf (2)

tεβ 1708.0 += (3)

cc Ef /2 '0 =ε (4)

where f ’c = concrete compressive strength; εt = tensile strain in normal direction to crack plane; ε0 = strain at peak stress; and Ec = Young’s modulus for concrete.

This model is expressed as the relationship between

principal stress and principal strain. Compressive strength is reduced according to the magnitude of tensile strain in the direction normal to the crack.

After peak stress, the effect of crack on compres-sion-softening is considered by the linear descending line (Fig. 7). In this model, compressive stress is reduced to zero at limited strain εu. However, the reduced stress has a limit that is 10% of the compressive strength. The gradient of strain softening is defined by the compressive fracture energy (Gfc) consumed in compressive stress parallel to the crack in the tension-compression area. The fracture energy for compression is determined by Na-kamura’s equation (Nakamura et al. 1999).

'88.0 cfc fG = (5)

The limit strain for compression (εu) is calculated from (Sato et al. 2004),

22 p

eqpeak

fcu l

G εσ

ε +⋅

= (6)

where εp = compressive strain at peak stress; σpeak = f ’c/β; and leq = equivalent length assumed to be the length of the element, 50 mm.

Shear transfer stresses were calculated using the model proposed by Li & Maekawa (1989). The smeared con-cept of considering concrete and steel reinforcement together as the reinforced concrete element is used in the finite element code so the bond between the concrete and steel reinforcement can be considered to have perfect rigidity.

4.2 Analyzed specimen Figure 8 shows the finite element mesh of a quarter model for saving calculation time and memory usage in the computation process. Fix conditions are given at the symmetry plane and loading points. For the XY plane of symmetry, only the Z-direction is fixed while for the YZ plane, the X-direction is fixed. The beams are simply supported beams subjected to two-point monotonic loading. In the analysis, the enforced displacements were

f ’c/β

ε

σ

ε0

f ’

εu 10%

εp Fig. 7 Compression model by fracture energy concept.


given at the loading points, which were the nodes of steel element attached to the specimen (Fig. 8). In the nu-merical results, softening of concrete in the concrete compression zone around the loading point in the flex-ure-shear region was observed at peak load. Thus the failure mode can be designated as shear compression failure.

5. Results and discussion

5.1 Load-deflection relationship and shear ca-pacity Figure 9 shows the experimental and computed load-deflection curves for specimens S1 and S2. It is very clear from the curve that the load-deflection relationship of the RC T-beam is changed because of the existence of the concrete top flange. The shear strength of the T-beam is also higher than that of the rectangular beam. The FE results show the same tendency. Moreover, it can be seen that the ultimate load is well predicted by the FE code. The difference of the initial stiffness between the nu-merical and test results in Fig. 9 can be observed. The reason for this difference may be the time dependent effects of concrete, the unexpected displacement at the supporting points, and the calculation algorithm by the modified Newton-Raphson method, which always tends to be stiffer than the actual response. Some examples of well-predicted results with the same FE program were also shown in previous work (Pansuk et al. 2004, 2006).

5.2 Crack patterns and failure mechanism The crack patterns of S1 and S2 are shown in Figs. 10 and 11, respectively. These figures show that the first diagonal shear cracks were initiated in each shear span of beams at angles 45o to 65o to the horizontal axis of the beam. Subsequently, additional diagonal cracks were formed with flatter angles as the load increased. The cracks extended up to the soffit of the flange in S2 (Fig. 11) and propagated into the direction of the connection zone between the flange and web. The effect of this horizontal crack in the T-beam on the shear resisting mechanism will be discussed later. At ultimate load, cracks were formed in the compression-zone concrete near the loading point, parallel to the top-fiber of the beam in the experiment. These cracks coalesced, result-ing in shear-compression failure of the compression- zone concrete for both tested specimens.

A similar crack propagation process can be obtained from the FE results. Moreover, the propagation of the horizontal crack in the analysis can be confirmed by the increment of strain in the vertical direction of the Gauss points near the connection zone of the flange and web at the load step corresponding to the test result, as shown in Fig. 12. The load drops after the shear crack reaches the compression zone of the beam in the analysis. This fail-ure process is observed in both analyses. A softening curve is observed in the computed normal compressive stress-strain or shear stress-strain relationship at the

Gauss points, where a crack occurred precisely around the loading point. The analytical failure process and crack pattern show that the numerical method in this study can satisfactorily evaluate the shear compression failure characteristic.

5.3 Stress development in specimens A typical pattern of the stress variation in stirrups for increasing loads was measured by the strain gages. The stresses plotted are the average for 2 locations of the stirrups. The location and reference number of stirrups

X

Y

Z

= RC element

= Concrete element

= Steel element

Restricted in plane

Fig. 8 Finite element mesh.

5 10 15 20

100

200

300

400

0Deflection (mm)

App

lied

load

(kN

) Test (S1) FEM (S1) Test (S2) FEM (S2)

Fig. 9 Load-deflection curves.

A (Test result at failure)

B (FE result at failure)

Fig. 10 Crack pattern of S1 (test and analysis).


whose strains were measured in the experiment are shown in Fig. 5. From Fig. 13, the average stresses of two selected stirrups from two locations (No. 1-2 and No. 4-5) were compared. Figure 13 shows that the average stresses of stirrups for both specimens are almost the same at the initial condition. After that, the average stress of the stirrup in the T-beam becomes lower than that in

the rectangular beam at around 250 kN of the applied load. At the same load level, correspondingly, the shear crack propagated horizontally below the concrete top flange at the same location of the considered locations (Figs. 5 and 11C). The lower stress of the stirrup in the T-beam can be considered to be due to the change in shear resisting mechanism, which will be discussed later.

In order to compare the test results with the smear-concept FE results, the average stresses of Gauss points from solid elements at the location corresponding to two considered stirrups were used. Analyzed and measured average stresses from locations between stir-rups No. 1-2 and No. 4-5 were compared as shown in Fig. 13. This figure shows that the FE results can predict the stress development in stirrup and the turning point of the shear resisting mechanism well.

A comparison between analyzed and measured aver-age strains in the direction parallel to the tension bars from the concrete strain gages placed in the compression area of both specimens is shown in Fig. 14. In order to compare the test results with the smear-concept FE re-sults, the average strains of Gauss points from solid elements and strain gages were used. The FE results can be said to be able to predict strain development in the compression area of both sections well (sections 1 and 2 are at the location of stirrup No. 1 and 5, respectively, see

A (Test result until 250 kN)

B (FE result until 250 kN)

Horizontal crack

C (Test result at failure)

Horizontal crack

D (FE result from 250 kN up to maximum load)

Fig. 11 Crack pattern of S2 (test and analysis).

0 1000 2000-0.002

-0.001

0

0.001

0.002

0.003

Distance from beam edge (mm)

Ver

tical

stra

in

Measured Gauss points

Fig. 12 Strain condition of element near connection zone between flange and web at 250 kN (not in scale).

100 200 300 400

100

200

300

400

0Stress in stirrup (MPa)

App

lied

load

(kN

) Turning point ofshear mechanism

Test (S1) Test (S2) FEM (S1) FEM (S2)

A (Location between stirrups no. 4 and 5).

100 200 300 400

100

200

300

400


App

lied

load

(kN

)

Test (S1)Test (S2)FEM (S1)FEM (S2)

Turning point ofshear mechanism

B (Location between stirrups no. 1 and 2).

Fig. 13 Stress development in stirrups (S1 and S2).


Figs. 5 and 14). Moreover, it can be observed from both the test and FE results that the average stress in the compression area of the T-beam is smaller than that in the rectangular beam due to the presence of a concrete top flange.

5.4 Variation of beam and arch action by stress distribution on sections Shear force (V) can be expressed by two components: arch action and beam action. At any location in a beam when a moment gradient dM/dx is present, these two effects are combined to give the total shear resistance. For a cracked concrete member, these components can be written as follows:

dxdMV = and jdTM ⋅= (7)

where T and jd are the tensile force in the bottom chord and lever arm, respectively. Thus

( )dxdTjd

dxjddTV += (8)

The first term of the previous equation refers to the arch action, while the second term describes the truss action. These two effects can be evaluated between two known sections along the length of the beam, thus Eq. (7) can be rewritten as

xTjd

xjdTV

∆∆

+∆∆

= (9)

By using the strain gage data in the concrete and ten-sion bars at sections of stirrup No. 1 and 5 (Fig. 15), the strain distribution on both sections can be drawn. By indicating the location of the neutral axis at different load steps and knowing the applied shear (V), all terms of Eq. (8) are known. Beam action variations for increasing shear force in both specimens were calculated from both the experimental results and the FE results between two sections and are shown in Fig. 16. Figure 16 shows that shear force starts out being carried entirely by the beam action but ends with the arch action being predominant

-0.002 -0.001 00

100

200

300

400

Concrete strain

App

lied

load

(kN

)

Test (Section 1)FEM (Section 1)Test (Section 2)FEM (Section 2)

A (Concrete strain in S1)

-0.002 -0.001 00

100

200

300

400

Concrete strain

App

lied

load

(kN

)

Test (Web)FEM (Flange)Test (Web)FEM (Flange)

B (Concrete strain in S2 (from section 1))

Fig. 14 Strain development in compression area of specimens.

-3000 -1500 0 1500 3000

100

200

300

Concrete strain

Bea

m h

eigh

t (m

m) 50 kN

100 kN 150 kN 164 kN

A (Specimen S1)

-3000 -1500 0 1500 3000

100

200

300

Concrete strain

Bea

m h

eigh

t (m

m)

50 kN 100 kN 150 kN 194 kN

B (Specimen S2)

Fig. 15 Strain distribution on sections.

50 100 150 200

50

100

150

200

0Applied shear force (kN)

Shea

r res

istin

g fo

rce

(kN

)

Total shear forceS1 (Test)S2 (Test)S1 (FEM)S2 (FEM)

Beam actionTurning point

Fig. 16 Shear resisting components.


for the T-beam. The load at which the beam and arch actions became different for two specimens corresponds to the turning point observed from the stress in stirrups. This result also shows the change of the governing re-sisting mechanism due to the presence of the concrete top flange. 6. Shear resisting mechanism of T-beam

6.1 Truss and arch action in T-beam From both the experimental and numerical results, it can be considered that the shear resisting mechanism of the T-beam is almost the same as that of the rectangular beam before the appearance of the horizontal crack. In this state, the beam action is governed by the truss mechanism. After the shear crack propagates horizon-tally below the concrete top flange, the stirrup stress of the T-beam becomes less than that of the rectangular beam. This can be considered to be due to the change of the governing shear resisting mechanism inside the beam from the truss mechanism to the arch mechanism. This is already confirmed by the comparison of shear resisting components (Fig. 16). In the arch mechanism, the con-crete area on the top flange of the T-beam can provide the additional area of the compression zone. 6.2 Change in governing mechanism due to horizontal crack From the experimental and numerical observation, the factor that controls the change of the governing mecha-nism of the T-beam from the truss mechanism to the arch mechanism is the propagation of the horizontal crack along the connection zone between the web and flange of the T-beam. The experimentally observed crack patterns indicate that the compression zone of the T-beam in-cluding the flange area can obstruct the continuous and instant propagation of diagonal cracks. Pimanmas and Maekawa (2001) reported that the diagonal cracks are unable to successfully penetrate across the plane with the low normal and shear traction transfer ability. The exis-tence of a top flange in the T-beam provides an additional resisting area so the discontinuity of average stress in the compression zone at the connection zone between the flange and web of the T-beam can be found. Because of this difference in average stress, the diversion of crack propagation occurs in the connection zone between the flange and web of the T-beam.

The change in governing mechanism due to horizontal cracking can be verified with 3D FEM. Numerical analysis of the specimen with bond-linkage elements was conducted to simulate the effect of the horizontal crack. This specimen had the same size and material properties as specimen S1.

The bond-linkage element used in this study is a layer element containing four Gauss points. In analysis, bond-linkage elements were installed at the position where the horizontal cracks were observed from the experiment using specimen S2 (along the connection

zone between the flange and web, see Fig. 17). The stiffness of the bond-linkage elements was initially set to the same value as the stiffness of concrete. Horizontal cracking was simulated by the sudden reduction of the shear transfer stress of bond-linkage elements at a given strain level in the vertical direction after shear crack. The shear transfer stresses were calculated using Li and Maekawa’s model (1998). From Fig. 18, after the shear transfer stress of bond-linkage elements is reduced (horizontal crack is assumed to have occurred in the analysis), the stress of stirrups in the beam with the horizontal crack truly becomes lower than that in the beam without such crack at the same load level. This confirms the influence of the horizontal crack as the governing factor for the shear mechanism of T-beams.

After the change in the governing shear resisting mechanism from truss action to arch action, it can be considered that the top part of the T-beam above the horizontal crack plays a very important role with regard to additional shear resistance. To confirm this hypothesis, the stress condition of elements in the compression zone at the location of stirrup No. 1 (Fig. 5) after the formation of the horizontal crack was investigated. It was found that the average principle stress of Gauss points from all elements above the horizontal crack increased for specimen S2 as shown in Fig. 19. The increment of av-erage principle stress after the horizontal crack is also observed for specimen S1 with a simulated horizontal crack (Fig. 19). The increase in resisting stress in the

Bond-linkage element

Gauss points

Install

Fig. 17 Bond-linkage element and installed location.

100 200 300 400

100

200

300

400

0

S1 (FEM) S1 (FEM)-with crack

Stress in stirrup (MPa)

App

lied

load

(kN

)

Fig. 18 Stirrup stress after simulated crack.


compression zone of specimen S1 with the artificial horizontal crack cannot improve the total shear capacity of this specimen (Fig. 20) because there is no additional resisting area from the concrete top flange to reduce the average stress, so the failure stage of the compression zone is reached before the arch action can be developed to as great an extent as in a normal T-beam. 6.3 Effect of flange width and thickness on the turning point To show the effect of flange width and thickness on the formation of the turning point and the increment of the shear strength of the T-beam through the arch mechanism, four additional numerical specimens, shown in Fig. 21, with a wider and thicker concrete top flange, were ana-lyzed. These specimens had the same material properties as specimens S1 and S2. Figure 22 shows that the in-crease in flange width has almost no additional effect on the characteristic of the horizontal crack, as shown by the strain condition of the element near the connection zone between the flange and web. Correspondingly, the turn-ing point of the governing mechanism for all T-beam specimens was observed at almost the same load level from the stirrup stress, as shown in Fig. 23. After the turning point, the average stress of the stirrup in the T-beam becomes lower than that in the rectangular beam, and the lowest average stirrup stress can be observed in

the T-beam with the widest top flange. Generation of a stronger arch flow in the T-beam with larger compression zone can be considered. This leads to the increase in shear strength of the T-beam after the flange width is enlarged, as shown in Fig. 24.

The numerical investigation shows that the horizontal crack cannot be observed in T-beam with the neutral axis located in the flange area (specimens T3 and T4, see Fig.

10 20 30 40 50

100

200

300

400

0Average stress (MPa)

App

lied

load

(kN

)

S1 (FEM) S1 (FEM)-with crack S2 (FEM)

Fig. 19 Average stress on compression zone.

5 10 15 20

100

200

300

400

0

S1 (FEM) S1 (FEM)-with crack

Deflection (mm)

App

lied

load

(kN

)

Fig. 20 Load-deflection curves after simulated crack.

75

T1

450

150

108

NA.

600

150

95

NA.

T2

150

T3

300

150

124 250

T4

300

140

Fig. 21 Addition numerical specimens (unit: mm).

0 1000 2000-0.002

-0.001

0

0.001

0.002

0.003


Ver

tical

stra

in

S1 S2 T1 T2


Fig. 22 Strain condition of element near connection zone between flange and web at 250 kN.

100 200 300 400

100

200

300

400


App

lied

load

(kN

)

S1 S2 T1 T2

Fig. 23 Comparison of stirrup stresses.


21) as shown by the strain condition of element near the connection zone between the flange and web (Fig. 25). Without the horizontal crack, the change of the govern-ing mechanism disturbing stirrup stresses cannot be observed, as shown in Fig. 26. A T-beam can be consid-ered the same as a rectangular beam of bf = bw if the neutral axis is located in the top flange zone. As a result, the ultimate strength of the T-beam is almost constant after the flange thickness is enlarged below the neutral axis location, as shown in Fig. 27.

6.4 Stress distribution and failure criteria of T-beam In order to predict the ultimate capacity of beams, the stress distribution on the beam section located closest to the loading point was investigated. Because of symmetry, the investigated Gauss points were chosen from only one half of the beam and were divided into a top layer and a second layer, as shown in Fig. 28. Figures 29, 30 and 31 show the distribution of the normal stress and shear stress along the top flange of the selected T-beam (T2) from the Gauss points shown in Fig. 28. Stresses are shown at three load levels: before ultimate load (300 kN), at the

10 20

100

200

300

400

0Deflection (mm)

App

lied

load

(kN

)

S1 S2 T1 T2

Fig. 24 Comparison of load-deflection curves.

0 1000 2000-0.002

-0.001

0

0.001

0.002

0.003


Ver

tical

stra

in

S1S2T3T4


Fig. 25 Strain condition of element near connection zone between flange and web at 250 kN.

100 200 300 400

100

200

300

400

0

S1S2T3T4

Stress in stirrup (MPa)

App

lied

load

(kN

)

Fig. 26 Comparison of stirrup stress.

10 20

100

200

300

400

0

S1S2T3T4

Deflection (mm)

App

lied

load

(kN

)

Fig. 27 Comparison of load-deflection curves.

Distance from

center

Top layer

Second layer

Fig. 28 Location of measured Gauss points.

0 100 200 300

20

30

Distance from center (mm)

Nor

mal

stre

ss (M

Pa) 300 kN

398 kN 390 kN

σx

Fig. 29 Distribution of normal compressive stress σx.


peak load (398 kN), and just after the peak load (390 kN). The distribution of stresses can be seen to change with the load level. After the turning point of the governing resisting mechanism, the development of stress (σy and τxy) in the flange zone decreases but a concentrated in-crease of stress can be observed from the zone at the center of the beam. Just after the peak load, softening of some Gauss points is observed followed by the redistri-bution of both normal and shear stresses to nearby Gauss points at the following calculation step. The redistribu-tion of stresses after the peak load can be considered to have no effect on the ultimate capacity of the beams and

the stress distribution at the ultimate state to be the most important predictor of the shear capacity of beams.

To simply evaluate the failure criteria of the compres-sion zone of T-beams with different flange widths, the shear stress distribution on the top flange of T-beams at failure load is considered, as shown in Fig. 32. The maximum shear stresses at failure can be seen to be al-most at the same level for all specimens. Moreover, the development of shear stress decreases with the distance from the beam center and becomes almost constant with a small value of shear stress at some distance. To define the value and distance at which shear stress becomes constant, observation of additional analytical specimens is necessary. Finally, the failure criteria of computed specimens can be shown by the relationship between the maximum and average stress on the top flange at failure load, as shown in Figs. 33, 34, 35 and 36. The maximum stress is averaged from Gauss points located inside web zone but the average stress is averaged from all Gauss points located in the top flange. Figure 33 shows that the maximum shear stresses at failure are almost at the same level for all specimens but the average shear stresses at failure become smaller with increases in flange width. This provides a clear explanation of the increase in shear strength of T-beams due to increases in top flange area. Moreover, the change of the slope of the relationship in Fig. 33 for T-beam specimens (specimens S2, T1 and T2) confirms the importance of the turning point of the shear resisting mechanism in T-beam. Figures 34 and 35 show that the maximum normal compressive stresses σx and σy at the failure zone are different for different top flange widths and the highest normal compressive stresses σx and σy at failure can be observed from rectangular specimen (S1 - without flange). The maximum com-pressive stress σx is greater than the concrete compres-sive strength (35 MPa). This can be explained by the confinement stresses acting on other directions of the failure zone. The relationship of normal compressive stress σz representing the confinement stress due to the top flange is shown in Fig. 36. Figure 36 shows that the

100 200 300

2

4

6

8

10

0Distance from center (mm)

Nor

mal

stre

ss (M

Pa) 300 kN

398 kN 390 kN

σy

Changes after

turning point

Fig. 30 Distribution of normal compressive stress σy.

100 200 300

2

4

6

8

10


Shea

r stre

ss (M

Pa) 300 kN

398 kN 390 kN

τxy

Changes after

turning point

Fig. 31 Distribution of shear stress τxy.

100 200 300

2

4

6

8

10

12


Shea

r stre

ss (M

Pa) S1

S2 T1 T2

End

of w

eb

Fig. 32 Comparison of shear stress distribution.

2 4 6 8 10 12

2

4

6

8

10

12


Max

imum

stre

ss (M

Pa)

S1 S2 T1 T2

Changes

after

turning

point

Fig. 33 Shear stress distribution at failure section.


confinement stress (σz) is smaller for the rectangular beam (S1). The maximum normal compressive stress σz at failure is almost at the same level for all T-beam specimens. Finally, it can be said that the exact failure criteria of the failure zone of the T-beam has to be con-sidered three-dimensionally. Moreover, the shear stress and the confinement stress (σz) at the ultimate state are

the main parameters indicating failure of the T-beam section.

From the stress distribution on the concrete top flange, the compression failure of concrete in the compression zone can be considered to take place in an area ap-proximately defined by the shaded part of the section in Fig. 37. The actual normal stress distribution (σx) is not uniform throughout the shaded area. The term “effective width” of a T-beam in shear can be defined as be = A/hf (Fig. 38), where A is the area of the shaded part of the cross section in Fig. 37, and hf is the thickness of the top flange below which the horizontal crack occurs. The average normal stress distribution is assumed to be uni-form throughout the “effective area.”

From the force equilibrium conditions, the ultimate capacity of beams is controlled by the smaller value between the compressive strength of the top strut (C) and the diagonal strut (N), as shown in Fig. 39. The failure of the top strut represents the shear compression failure mode, and the failure of the diagonal strut represents the web crushing failure mode. The strength of the top strut can be calculated based on the failure criteria expressed by the relationship between the maximum and average stresses on the top flange at failure in this study. Further analysis of these systems will be the subject of continu-ing work.

10 20 30 40 50

10

20

30

40

50


Max

imum

stre

ss (M

Pa)

S1 S2 T1 T2

Fig. 34 Normal compressive stress (σx) distribution at failure section.

2 4 6 8 10

2

4

6

8

10


Max

imum

stre

ss (M

Pa)

S1 S2 T1 T2

Fig. 35 Normal compressive stress (σy) distribution at failure section.

1 2 3 4 5 6

1

2

3

4

5

6


Max

imum

stre

ss (M

Pa) S1

S2 T1 T2

Fig. 36 Normal compressive stress (σz) distribution at failure section.

bf / 2

hf

Fig. 37 Cross section of T-beam with effective area in shear (shaded area = A).

be = A/hf

hf

Fig. 38 Effective width of T-beam in shear.


7. Conclusions

The following conclusions can be drawn based on the findings of this study: (1) Increases in the flange width of a T-beam give higher

shear capacity with a nonlinear relationship for the T-beam with shear reinforcement. In the case of a T-beam without shear reinforcement, however, the width of the flange has almost no effect on shear capacity.

(2) The existence of a concrete top flange has a signifi-cant effect on the shear behaviors of RC T-beams. In view of the good correlation between the nonlinear finite element analysis and the experimental results, it would seem that these effects can be simulated well by the finite element code (CAMUI).

(3) The governing shear resisting mechanism of a T-beam is changed from the truss mechanism to the arch mechanism because of the formation of a hori-zontal crack along the boundary between the top flange and the web. After the turning point of the shear resisting mechanism, the shear behaviors of RC T-beams such as stirrup stress and concrete stress in the compression zone are different from those of rectangular beams.

(4) In the arch mechanism, the concrete area on the top flange of a T-beam can provide an additional area for the compression zone so the average stress at failure can be reduced and the shear capacity of the T-beam can be improved.

(5) Increases in flange width have no effect on the for-mation of the turning point. On the other hand, the inclusion of a flange increases the ultimate shear strength, but only down to a certain width. Above this width, the effect of the flange area on the shear strength is less significant. A full investigation on the effect of the geometrical parameters on shear capac-ity is to be the subject of further research.

(6) A simplified method for determining the failure cri-teria of the compression zone of a T-beam for shear is proposed. Accompanied by the force equilibrium conditions, a simplified concept for the evaluation of the shear strength of a T-beam can be proposed. Verification of this simplified method will be the subject of continuing work.

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C

TN

V

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Fig. 39 Strength of concrete strut for a concentrated load near a support.


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Shear Mechanism of Reinforced Concrete T-Beams with Stirrups

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