Indian Journal of Engineering & Materials Sciences Vol. 25, December 2018, pp. 487-497

Flexural analysis of posttensioned reinforced concrete beams using external steel bars

Swoo-Heon Leea, Hee-Du Leeb & Kyung-Jae Shinb* aSchool of Convergence & Fusion System Engineering, Kyungpook National University, Gyeongsangbuk-do37224, South Korea

bSchool of Architectural Engineering, Kyungpook National University, Daegu 41566, South Korea

Received 8 January 2016; accepted 14 November 2017

A nonlinear analysis based on the incremental deformation method has been presented to predict the flexural response of reinforced concrete (RC) beams strengthened with externally unbonded steelbars instead of tendons. The steel bars (with ultimate strength between 668 and 788 MPa) generally have lower strength than tendons (with ultimate strength between 1400 and 1900 MPa), and can be posttensioned simply by tightening nuts or turnbuckles, unlike a tendon. The load–deflection, load–external bar stress and moment–curvature over the entire loading range are examined to evaluate the flexural response of RC beams with or without external steel bars. To satisfy force equilibrium, iterative processes for obtaining the neutral axis and external bar stress in the cross-section are performed. This iterative procedure with nonlinear analysis is essential because there is an interrelationship between the neutral axis, external bar stress, and beam deflection. The nonlinear analytical results for the deflection and external bar stress show good agreement with experimental values. Additionally, deformed shapes such as varying neutral axis and curvature distribution along the beam span, which were not observed in experimental tests, are also exhibited.

Keywords: Reinforced concrete beam, Strengthening, External steel bar, Nonlinear analysis, Posttensioning bar, Flexural behavior

External posttensioning by tendons has been used in flexural members to strengthen or retrofit structures for many years. Numerous experimental and analytical papers on external prestressing by steel tendons have been published. Alkhairi and Naaman1

proposed an iterative nonlinear analysis for prestressed beams with internal or external tendons. The basic assumptions and analysis model based on nonlinear analysis reported by Alkhairi and Naaman1

became the foundation for the work performed by Harajli et al.2 and Yang and Kang3. The iterative procedure was performed to satisfy force equilibrium at the cross-section and deformation compatibility along the member length or at the deviator.

Tan and Ng4 carried out experimental and analytical research on prestressed T-beams. The effect of tendon configuration due to the presence or absence of deviator(s) and prestressing level on the prestressed beam was described. The deviator(s) minimized the second-order effect due to eccentricity, and the bond reduction factor was used to evaluate the experimental results for ultimate strength, corresponding deflection, and tendon stress.

Harajli et al.2 presented a nonlinear analysis based on incremental deformation to demonstrate the second-order effect of external tendons. The model has been verified through experimental results for simply supported T-beams as reported by Tan and Ng4 and Khairallah5.

Over the last decade, several new analysis models have been developed to evaluate the response of prestressed beams using tendons. Wu and Lu6

simulated tendon slip and friction on the deviator(s) for a few externally prestressed beams reported by Harajli et al.2 and Tan and Ng4. Dall’Asta et al.7

provided a model to describe geometric nonlinearity by the kinematics of a slipping coupled system comprising a beam and tendon. Dall’Asta et al.8 developed a simplified method without nonlinear analysis to find the incremental stress of tendons by using shape functions. The analytical articles of Pisani9 and Lou et al.10 focused on prestressed continuous concrete beams. Lou and Xiang11 also proposed a finite element model for nonlinear analysis of prestressed simply supported beams with steel tendons. Yang and Kang3 determined the equivalent strain distribution factor by comparing numerous test data for simply supported beams with a 2-D nonlinear

———————— Corresponding author (E-mail: shin@knu.ac.kr)

INDIAN J. ENG. MATER. SCI., DECEMBER 2018

488

analysis based on the analysis method of Alkhairi and Naaman1.

Other researchers have worked with different materials used for posttensioning (e.g., Parafil rope, paper-wrapped tendon, or steel bar). Ghalla and Beeby12 carried out tests on beams with Parafil rope and performed analysis for stress in steel tendons as well as Parafil rope. Bondy13 introduced external posttensioning using paper-wrapped tendons for repair work of structures damaged due to corrosion. Shin and Lee14 conducted experimental work on seven beams strengthened by draped steel bars and three controlled reinforced concrete beams.

In this paper, the test data of Shin and Lee14, which used steel bars instead of tendons as strengthening material, are examined within the scope of this analytical study. Some information from cited literature is summarized in the following section. Strain compatibility between the posttensioning bars and surrounding concrete is inapplicable to the analysis of externally posttensioned concrete beams because the behavior of external bars is not section dependent using strain compatibility but member dependent. Numerical work considering member-dependent characteristics was carried out to better understand the posttensioned concrete beams. The analytical results matched well with the experimental results and included unobservable findings in the tests. Overview of Previous Experiment

Shin and Lee14 performed experimental testing for ten full-scaled reinforced concrete beams with a section of 400 mm × 600 mm and a span length of 6,000 mm. In this experiment, one-point loading at the mid-span was applied, and the external bar system showed a V-shaped profile from one deviator

installed at the mid-span. The variables are summarized in Fig. 1 and Table 1. The amount of tension reinforcement (As = 1935, 2709, and 3549 mm2) and diameters of the external bar (18, 22, and 28 mm) were used as variables. The friction on the contact surface between the deviator and beam bottom, and the force loss due to local deformation at the anchorage pin, are not considered in the analysis of the next section. In Fig. 1 and Table 1, A's and As are the area of compression and tension reinforcement, respectively; f'y and fy are the yield strength of compression and tension reinforcement, respectively; f'u and fu are the ultimate strength of compression and tension reinforcement, respectively; Ap is the area of the external steel bar; fpy and fpu are the yield (obtained by the 0.2% offset method) and ultimate strength of the external steel bar, respectively; fse is the effective stress in the external steel bar after relaxation; fcm is the average compressive strength of concrete cylinders at 28 days; b is the beam width; d is the effective depth of the beam; dp is the distance from the beam top to the center of the deviator at the mid-span; and d' is the distance from the extreme compressive fiber to the centroid of the compression steel bar. Nonlinear Analysis Model

Basic assumptions The nonlinear analysis in this study was based on

the following assumptions: (i) Plane sections remain plane before and after

bending deformation; (ii) The internal reinforcement is perfectly bonded to

the concrete, so the reinforcement strain is equal to that of the surrounding concrete;

(iii) The flexural strain in the beam cross-section is a straight line from the extreme compression fiber to the internal tension reinforcement;

Fig. 1 — Concrete beam tested by Shin and Lee14

LEE et al.: FLEXURAL ANALYSIS OF POSTTENSIONED REINFORCED CONCRETE BEAMS

489

(iv) Shear deformations are negligible due to sufficient transverse reinforcement;

(v) Loading pattern, external bar configuration, and beam deformation (i.e., slope and deflection) are geometrically symmetrical;

(vi) The elongation of the external steel system occurs only by deflection at the deviator;

(vii) The strain and/or stress of external steel bar is constant due to uniaxial deformation between the deviator and anchorage;

(viii) The force loss produced by friction at the deviator is not considered; and

(ix) The bending flexibility of the end anchorage is neglected.

Stress–strain relationship of materials

This study refers to the investigation of Lou et al.10

and Hognestad15 on the stress–strain relationship of concrete as illustrated in Fig. 2. The compressive behavior of concrete exhibits a parabolic ascending branch with increased strain up to 0 and a linear descending branch with increased strain up to cu. The maximum compressive strain of concrete cu is based on values measured using crack gauges in previous experimental literature, but if its value was uncertain,

it was assumed to be 0.005 [which is taken for rehabilitation based on previous research by Paulay and Priestley16 and FEMA17]. The tensile behavior of concrete is assumed to be elastic-linear up to a cracking strength ft. Beyond ft, the concrete tensile stress is decreased linearly to zero at tu. The stress fc and strain c in concrete is as follows:

For 0 ≤c≤0

2

00

2

cc

cmc ff … (1a)

Table 1 — Summary of specimens reported by Shin and Lee14

Specimen Reinforcements External steel bars Concrete

A's (mm2)

f'y/ f'u (MPa)

As (mm2)

fy/ fu (MPa)

Ap (mm2)

fpy/fpu (MPa)

fse (MPa)

dp (mm)

No. of deviators

fcm (MPa)

RC-C

860 (3-D19)

335 / 506

1936 (5-D22)

434 / 605

- - - - -

24.3

T18V-LC

860 (3-D19)

335 / 506

1936 (5-D22)

434 / 605

508 (2-18)

642 / 788

248

634

1 (at midspan)

24.3

T22V-LC

860 (3-D19)

335 / 506

1936 (5-D22)

434 / 605

760 (2-22)

584 / 776

219

634

1 (at midspan)

24.3

T28V-LC

860 (3-D19)

335 / 506

1936 (5-D22)

434 / 605

1232 (2-28)

562 / 668

198

634

1 (at midspan)

24.3

RC-C-S1

860 (3-D19)

335 / 506

2710 (7-D22)

434 / 605

- - - - -

24.3

T18V-LC-S1

860 (3-D19)

335 / 506

2710 (7-D22)

434 / 605

508 (2-18)

642 / 788

243

634

1 (at midspan)

24.3

T28V-LC-S1

860 (3-D19)

335 / 506

2710 (7-D22)

434 / 605

1232 (2-28)

562 / 668

212

634

1 (at midspan)

24.3

RC-C-S2

860 (3-D19)

335 / 506

3547 (7-D25)

398 / 613

- - - - -

24.3

T18V-LC-S2

860 (3-D19)

335 / 506

3547 (7-D25)

398 / 613

508 (2-18)

642 / 788

197

634

1 (at midspan)

24.3

T28V-LC-S2

860 (3-D19)

335 / 506

3547 (7-D25)

398 / 613

1232 (2-28)

562 / 668

208

634

1 (at midspan)

24.3

Fig. 2 — Idealized stress–strain curve of concrete

INDIAN J. ENG. MATER. SCI., DECEMBER 2018

490

And for0<c≤cu

0

015.01

cu

ccmc ff … (1b)

where fcm is the mean value of compressive strength of the concrete cylinder; cu is the maximum compressive strain; 0 is the strain at peak stress, taken as 0.002; ft is tensile strength of the concrete, taken as 0.31 fcm

0.5 (MPa); tu is the ultimate tensile strain, taken as 10ft/Ec; and Ec is the elastic modulus of concrete, estimated as 4700 fcm

0.5 (MPa) by the American Concrete Institute (ACI) 318-1118. Actual stress–strain curves taken from the tensile tests for reinforcement and external steel were used as shown in Fig. 3. Analysis procedure

The cross-section on which the moment–curvature analysis will be performed is determined in advance.

The detailed section information (e.g., beam section size, areas and level of reinforcement and external steel, material properties, etc.) is setup, and a concrete section is divided into n laminas and three steel layers [first layer (n + 1) for the compression reinforcement, second layer (n + 2) for the tension reinforcement, and third layer (n + 3) for the external steel bar] are added for a rectangular section [see Fig.4(a)]. A force equilibrium analysis is initiated by increasing the curvature (i,j) for the cross-section. The strain of every lamina is proportional to the distance from the temporarily assumed neutral axis dN as indicated in Figs 4(b) and 4(c). In addition, the concrete strains due to effective prestrain of the external steel and the strain in the external steel produced by incremental vertical deflection are considered. Compared with an RC beam without strengthening of external posttensioned steel bars, the RC beam strengthened with a draped steel bar has different cross-sections along the beam length as shown in Figs 1 and 5. The explanation below is focused on strengthening of the tensioned member, but it is available for RC beams with no external steel if the components of the external steel bar are neglected.

In this stage, the force equilibrium process at the critical cross-section (corresponding to the 1st small beam element in Fig. 5) can be traced numerically as follows. Analysis of the remaining small beam elements follows the same procedure.

Step 1: Specify the curvature (i,1) at the i-th loading stage.

Step 2: Assume a neutral axis depth dN at the cross-section.

Fig. 3 — Stress–strain relationship for steel materials

Fig. 4 — (a) Lamina discretization (k=1~n + 3), (b) linear distribution of strain, (c) idealized strain and (d) internal forces in cross-section of 1st beam element at i-th loading stage

LEE et al.: FLEXURAL ANALYSIS OF POSTTENSIONED REINFORCED CONCRETE BEAMS

491

Step 3: Calculate the strain distribution of every lamina by using their distances from the assumed neutral axis dN to every lamina and the incremental curvature. Herein, the strain of the externally unbonded steel at the i-th loading is unknown because it is impossible to obtain its value from strain compatibility. Therefore, the increased strain of external steel is assumed as a guess value. After calculating the beam deflection, the strain of externally unbonded steel should converge.

),1()1,(),1,( kyiki … (2)

where i indicates a loading stage number in sequence; k indicates the layer number of the divided lamina from top to bottom (from 1 at the top to n at the bottom plus three layers for steel); and y(1,k) is the distance from the assumed neutral axis to each lamina.

Step 4: Find stresses acting on each lamina and steel using linear interpolation from the stress–strain curve of the materials.

Step 5: Calculate the internal forces for every lamina and steel from

)(),1,(),1,( kAkifkiN c … (3a)

sss AifiCniN )1,()1,()1,1,( … (3b)

sss AifiTniN )1,()1,()2,1,( … (3c)

ppp AifiTniN )1,()1,()3,1,( … (3d)

where A(k) is the area of the lamina; A's and As are the areas of the compression and tension steel reinforcements, respectively; Ap is the area of the

external steel; fc(i,1,k) is the stress on the concrete lamina; f's(i,1) is the stress on the compression reinforcement; fs(i,1) is the stress on the tension reinforcement; fp(i,1) is the stress on the external steel bar and is constant at all small beam elements [i.e., fp(i,1)=fp(i,j)]; and j is the element number of the subdivided small beam (from 1 at the mid-span to m at the support).

Step 6: Check whether the internal force is in equilibrium, and if not, reassume a temporary neutral axis dN at Step 2 by using the bisection method and continue iteratively until the force equilibrium is satisfied. The axial force of Eq. (4) acts on the beam by the posttensioning of external steel and/or deformation of external steel due to the beam deflection. If the internal forces are in equilibrium, the temporary dN is referred to as the specified neutral axis of ӯ(i,1) at the i-th loading stage.

3

1

force Axial),1,(n

k

kiN … (4)

Step 7: If force equilibrium at the critical section is achieved, the internal moment capacity at the cross-section can be obtained by Eq.(5). The moment is computed by summing the internal moments of every component with respect to the specified neutral axis.

∑3

1

),1(),1,()1,(

n

k

kykiNiM … (5)

where y(1,k) is the distance from the specified neutral axis ӯ (i,1) to each lamina.

Fig. 5 — (a) Subdivided small beam elements (j= 1~m) along the length and (b) strain and internal forces at 1st and j-th elements

INDIAN J. ENG. MATER. SCI., DECEMBER 2018

492

Step 8: Finally, a moment–curvature relationship [M(i,1) vs. (i,1)] is established at the critical cross-section at the i-th loading stage. The applied load of P(i) is calculated according to the loading pattern.

LiMiP /)1,()( … (6)

where is the coefficient according to the loading pattern (e.g.,=4 for a single-point loading, =6 for two equal 1/3-point loadings, and 8/L for a uniformly distributed loading).

After specifying the moment–curvature relationship of the critical cross-section, the beam from the mid-span to the end support is subdivided into small beam elements of m as shown in Fig. 5(a).

Step 9: The bending moment Me(i,j) at every element is proportionally calculated from M(i,1) of Eq.(5) at the critical section by using the bending moment diagrams. J is the element number, which is determined so as to approach the rational result (from 1 at the mid-span to m at the end support).

Step 10: As shown in Fig. 5 and mentioned earlier, the cross-section of each beam element along the beam length is different because the external steel was installed from the beam bottom (at mid-span) to the beam top (at support). Therefore, the moment–curvature analysis for all varying cross-sections along the beam length should be performed through the procedure outlined in Steps 1 to 7. To satisfy the force equilibrium of Eq. (4) and Me(i,j) of Step 9, the curvature (i,j) and the neural axis depth ӯ(i, j) for every small beam element should be converged. In this step, the stress/strain of the external steel in all remaining small elements should be fixed as the value assumed during critical-section analysis. It should be noted that the external bar stress is constant along the beam length.

Step 11: The rotation and deflection of a beam member can be calculated by integrating the curvatures along the beam length according to Park and Paulay19. Because the curvature is defined as the rotation per unit length of a member, the rotation angle in the j-th element at the i-th loading stage is given by

dxjijij

j

∑1

),(),(

… (7)

where dx is the length of a small element.

Step 12: The deflection at the mid-span is given by

∑m

jspanmid dxjii

1

),()(

… (8)

Step 13:To obtain the strain ofthe external steel, the deflection at the deviator should be calculated. If the deviator is in the mid-span, Eq.(9) should be replaced with Eq.(8). Even if there are several deviators, each deflection can still be calculated.

∑m

deviatorjdeviator dxjii

1

),()(

… (9)

Step 14: The incremental elongation p(i) of the external steel due to the deflection (mi-span or deviator) can be calculated from Eqs (10)- 12) based on the geometry of Fig. 6 before and after deformation. This incremental strain p(i) should be added to the effective pre-strain se [that is, p(i)=se + p(i)], and the stress fp(i) can be found by linear interpolation from the stress–strain curve of external steel.

pppp LLL /)-( … (10)

22 ]-[]-2/[( vphp dddLL … (11)

22 ]-[])()2/[( deviatorvpvhp dddy--dLL … (12)

where p is the incremental strain of the external steel; Lp is the external steel length before deformation; L'p is the external steel length after deformation; dh is the horizontal distance from anchorage to support; dv is the vertical distance from anchorage to beam top; ӯ is the neutral axis depth; is the rotation angle at anchorage; dp is the effective depth of the external steel at the deviator; and deviatoris the deflection at the deviator point.

Step 15: Here, the tolerance between the initial guess value for the strain of the external steel in Step 3 and the external steel strain due to deflection in Step 14 should be reduced. By reassuming an initial guess using the bisection method, the iterative

Fig. 6 — Deformation of external steel and beam

LEE et al.: FLEXURAL ANALYSIS OF POSTTENSIONED REINFORCED CONCRETE BEAMS

493

solution from Step 3 to Step 14 is repeated until the absolute tolerance is within the1% selected for this study.

Step 16: When satisfying the tolerance, the applied load (or moment) and deflection at the mid-span and/or at the deviator are determined. The iteration process then returns to Step 1. This analysis process continues until the strain in the extreme compressive fiber reaches cu that was mentioned in Section 3.2. The procedure utilizing nonlinear analysis is provided in the flowchart of Fig.7. Verification

Using the nonlinear analysis described in the above section, the load–deflection curve (or moment–curvature curve) at the mid-span and the load–external bar stress curve are obtained and plotted in

Figs 8-14. The comparison of predicted and measured results of load for various beams is given in Table 2. Load–deflection and moment–curvature relationships

In the analytical deflection curves, the minor uplift deformation of negative values due to posttensioning was not considered. Figure 8 compares the predicted

Fig. 7 — Flowchart for nonlinear analysis of externallyposttensioned beam at i-th loading stage

Fig. 8 — Predicted and measured load–deflection at mid-span

Fig. 9 — Predicted and measured moment–curvature at critical cross-section

Fig. 10 — Predicted and measured load–external bar stress

INDIAN J. ENG. MATER. SCI., DECEMBER 2018

494

and measured load–deflection at the mid-span of the beams. The analytical approach for predicting the load–deflection matched well with the experimental results up to yielding. In each curve, the elastic behavior was plotted as the first stage, and then the post cracking behavior with crack occurrence in the bottom fiber was continued up to yielding as the second stage. As the third stage, the load and deflection were increased until the end of the analysis. However, when the concrete strain in the extreme top fiber reached the measured compressive strain, the deflection (at the marked diamond point) did not reach the measured deflection after yielding. A

potential reason for this could be that concrete crushing failure did not occur at the location where the crack gauges (see Fig. 1) were installed, so the actual maximum compressive strain at ultimate strength might have been much greater than the measured value εcu (Shin et al.20). All measured strains cu in the tests were greater than 0.005, which can be taken for rehabilitation. When cu was assumed to be 0.005 or measured strain, the analytical deflection and the stress of external bars did not reach the values measured in the experiments. To reach the measured deflection, maximal compressive strains cu greater than 0.03 in the top fiber were required: they were approximately 0.03 for models without external steel and 0.05 for models with external steel. Because the beam T18V-LC failed during fracture in the threaded part of the external steel during the loading test, its test was terminated at a deflection of approximately 64 mm.

Similarly, the moment–curvature had three stages. The moment capacity also increased significantly with increasing amount of reinforcement. The cracking moments (or loads) exhibited similar capacities because the elastic behavior of the uncracked section was mostly governed by the concrete section rather than by the effect of the reinforcement ratio or added external bars.

Fig. 12 — Stress variation for embedded and external steel bars at (a) T22V-LC, (b) T28V-LC, (c) T18V-LC-S1, and (d) T28V-LC-S2

Fig. 11 — Stress in embedded steel bars with increased load

LEE et al.: FLEXURAL ANALYSIS OF POSTTENSIONED REINFORCED CONCRETE BEAMS

495

The predicted yielding moments of the models strengthened with external bars were underestimated slightly. The analytical ultimate moment capacities are analogous to the measured ones, but the analytical curvatures are much greater than the measured ones. The reason may be that the curvatures were obtained from strains measured using crack gauges, and these crack gauge strains were measured smaller than the actual strains. Stress in external bars and embedded steelbars

In Fig.10, the numerical behavior (dashed line) and experimental behavior (solid line) are compared for applied load–external bar stress. Good agreement between the measured and analytical models is

Table 2 — Comparison of predicted and measured results

Specimen At yielding At ultimate

Py_mea (kN) Py_pre (kN) Py_mea/Py_ pre Pu_mea (kN) Pu_pre (kN) Pu_mea/Pu_ pre

RC-C 341 321 1.06 395 413 0.96

T18V-LC 418 401 1.04 469 465 1.01

T22V-LC 441 428 1.03 546 580 0.94

T28V-LC 519 480 1.08 628 649 0.97

RC-C-S1 447 442 1.01 491 537 0.91

T18V-LC-S1 544 511 1.06 618 621 1.00

T28V-LC-S1 634 594 1.07 685 716 0.96

RC-C-S2 539 533 1.01 614 670 0.92

T18V-LC-S2 631 623 1.01 691 752 0.92

T28V-LC-S2 687 676 1.02 751 804 0.92

Average 1.04 0.95

Standard deviation 0.03 0.03

Fig. 13 — Actual crack pattern and neutral axis variation

Fig. 14 — Curvature distribution along the beam span

INDIAN J. ENG. MATER. SCI., DECEMBER 2018

496

shown. In only two beams, T22V-LC and T28V-LC, the external bar stress reached its yield strength. These two beams have relatively lower amounts of internal reinforcement. As the amount of reinforcement, including both internal and external steel bars, was increased, the ductility (or deflection) and external bar stress decreased together owing to the interrelationship of deflection and bar stress. However, the ordinary reinforcements, such as compressive and tensile steel bars, yielded and/or hardened in all models.

Stresses of the embedded steel bars at the mid-span with increasing load are shown in Fig. 11. In the RC-C model without external bars, the stress increase in the bottom reinforcement is greater than that in the top one with increased load, and the tension bar yields first at approximately 300 kN. Thereafter, the tension steel bar was hardened, and the stress in the compression steel bar was rapidly increased to its yield stress. The initial stress increases in the compression steel bars up to an applied load of 300 kN were mostly similar in all models, but those in the tension reinforcement were reduced with higher amounts of steel, including internal and external steel bars. Both compression and tension reinforcements yielded at nearly the same time in all models.

Figure 12 presents the varying stress of internal steel bars (at the mid-span) and external steel bars with increased load, in four representative models. The two models of T22V-LC and T28V-LC were the only models in which the external bars yielded at the ultimate state, but all the internal steel had reached the yielding state already. As the applied load increased, the increases in the external bar stress was more obvious than that of the internal steel until the applied load reached the yielding load of the external bar beam. Thereafter, the rise of the external bar stress progressed very slowly, but the internal steel stress increased gradually until ultimate load. Neutral axis depth

Figure 13 shows the ultimate varying neutral axis depth and crack patterns along the beam span (or at the end of analysis). Comparing the three models without external bars (of RC-C, RC-C-S1 and RC-C-S2), the neutral axis of model RC-C-S2 with larger reinforcement is smaller than the other two models. From the mid-span to 700 mm for the RC-C model, the higher neutral axis is exhibited because there are few main flexural cracks with large widths. Similarly, the RC-C-S1 and RC-C-S2

models have this tendency from the mid-span to 600 mm. This zone would be the plastic hinge zone according to Park and Paulay19. When the ultimate strength of a member is reached, this plastic zone plays an important role in determining the plastic deformation. In the region after the plastic hinge to 2,000 mm, the neutral axis depth is constant, indicating stabilized cracking development in the three models with no external bars. Hereafter, the neutral axis depth of the gross-area section with few or no cracks begins to decrease.

The neutral axis depths in the models with external bars are lower than those in the models without strengthening. This can be explained as the added external bars playing the role of tension reinforcement. In addition, the plastic hinge range is much shorter than in models with no external bars. The neutral axis begins to lie outside the cross-section at distances greater than 2,000 mm from the support. A reasonable explanation is that the external bars are located above the center of the cross-section from approximately 2,000 mm and the axial force from the external bars heavily influences the concrete section, more so than flexure from the main loading. Curvature distribution

Figure 14 shows the curvature distribution along the beam span at ultimate load. The curvature is greater around the mid-span area where the location of the neutral axis is high as shown in Fig. 13, but it is relatively small and decreases linearly in the remaining zone. Numerous cracks developed at the mid-span, whereas small cracks or no cracks are visible in the remaining zone. The crack patterns of T22V-LC and T18V-LC-S1 are also similar to those of the three models with external bars in Fig. 13. As the reinforcement ratio and/or external rod diameter increases, the extent of the larger curvature zone decreases, and the maximal curvature is also reduced. The extent of the larger curvature zone is the plastic hinge length (lp). Inelastic rotation occurs at this zone. The ultimate deflections are proportional to the areas of curvature. The additional strengthening by external bars leads to local concrete crushing and more flexural cracks at the mid-span, but similar curvature remained at the other zones. In addition, the more external steel bars in the beam, the shorter the plastic hinge length. This results in less deflection or ductility. lp was greater than d =540 mm in the models without external bars but was less than d in the posttensioned beams.

LEE et al.: FLEXURAL ANALYSIS OF POSTTENSIONED REINFORCED CONCRETE BEAMS

497

Conclusions In this study, a nonlinear approach by the lamina

section method was utilized to investigate the entire flexural response for posttensioned reinforced concrete beams. This paper focuses on concrete beams strengthened by steel bars in place of tendons (which is one of the most commonly used materials in the construction field). The following conclusions are drawn based on the nonlinear analysis results: (i) The method proposed in this paper matches well

with the incremental deflection, stress of the external steel bars, and curvature with respect to the applied load (or moment), although the friction and local deformation of external steel bars at the deviator and anchorages were neglected.

(ii) With increased reinforcement and external bars, RC beams exhibited higher load capacity (i.e., higher yielding and ultimate loads) and a decrease in ductility.

(iii) In addition, the proposed method can predict the varying curvature distributions and neutral axis depth along the beam span, which were not measured in previous experimental works. In the model with external bars as strengthening material, the neutral axis around the supports may lie outside the cross-section of concrete owing to the axial force from the external bars.

(iv) The extent of the curvature zone (= plastic hinge length) became smaller with increasing amount of internal reinforcement and added external bars. For this reason, the models with larger amounts of reinforcement have smaller deflections and ductility.

Acknowledgement

This research was supported by the Kyungpook National University Research Fund.

References 1 Alkhairi F M & Naaman A E, J Struct Eng, ASCE, 119(9)

(1993) 2680-2700. 2 Harajli M, Khairallah N & NassifH, J Struct Eng, ASCE,

125(10) (1999) 1151-1161. 3 Yang K-H & Kang T H-K, ACI Struct J, 108(2) (2011)

217-226. 4 Tan K-H & Ng C-K, ACI Struct J, 94(1) (1997) 13-21. 5 Khairallah N E, Experimental and Analytical Evaluation of

the Behavior of Concrete T Beamswith External Prestressing, MS thesis, American University of Beirut, Beirut, 1997.

6 WuX-H & Lu X, J Struct Eng, ASCE, 129(1) (2003) 96-104. 7 Dall’Asta A, Ragni L & Zona A, J Eng Mech, ASCE, 133(1)

(2007) 117-121. 8 Dall’Asta A, Ragni L & Zona A, J Struct Eng, ASCE, 133(1)

(2007) 121-131. 9 Pisani M A, J Bridge Eng, ASCE, 14(2) (2009) 93-101. 10 LouT, Lopes S M R & Lopes A V, J Bridge Eng, ASCE,

18(6) (2013) 525-537. 11 Lou T & Xiang Y, Eng Struct, 28(14) (2006) 1919-1926. 12 Ghallab A & Beeby A W, Cem Concr Compos, 27(9-10)

(2005) 945-957. 13 Bondy K D, PTI J, 4(2) (2006) 24-29. 14 Shin K-J & Lee S-H, Mag Concr Res, 62(2), (2010) 137-147. 15 Hognestad E, A study of Combined Bending and Axial Load

in Reinforced Concrete Members (University of Illinois Engineering Experimental Station, Illinois), 1951.

16 Paulay T & Priestley M J N, Seismic Design of Reinforced Concrete and Masonry Buildings (John Wiley & Sons, New York), 1992.

17 FEMA 274, NEHRP Commentary on the Guidelines for the Seismic Rehabilitation of Buildings (Federal Emergency Management Agency, Washington D.C.), 1997.

18 ACI Committee 318, Building Code Requirements for Structural Concrete (ACI 318-11) and Commentary (ACI 318R-11) (American Concrete Institute, MI) 2011.

19 Park R & Paulay T, Reinforced Concrete Structures (John Wiley & Sons, New York), 1975.

20 Shin K-J, Lee S-H & Kang T H-K, J Struct Eng, ASCE, 140 (3) (2014) 04013067.