In-plane flexural strength of unbonded post-tensioned concrete masonry walls

Engineering Structures 136 (2017) 245–260

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In-plane flexural strength of unbonded post-tensioned concrete masonry walls Reza Hassanli a, Mohamed A. ElGawady b,⇑, Julie E. Mills a a b

University of South Australia, Adelaide, Australia Missouri University of Science and Technology, MO, USA

a r t i c l e

i n f o

Article history: Received 7 January 2016 Revised 3 January 2017 Accepted 5 January 2017

Keywords: Masonry MSJC2013 Shear strength Flexural strength Unbonded post-tensioning bars In-plane behavior, post-tensioned wall

a b s t r a c t The target of this paper is to develop a design equation to predict the in-plane flexural strength of unbonded post-tensioned masonry walls (PT-MWs). Using validated finite element models, a parametric study was performed to investigate the effect of different parameters on the wall rotation and compression zone length. Multivariate regression analysis was performed to develop an equation to estimate the rotation of the unbonded PT-MWs at peak strength. Using the drift capacity of the walls and the proposed equation, a design expression and a related step-by-step design method have been developed to estimate the flexural strength of unbonded PT-MWs. The accuracy of the procedure was examined using experimental and finite element model results. Ignoring the elongation of the PT bars in the strength prediction resulted in an underestimation of about 40%, while using the proposed approach the prediction improved significantly. It was found that the wall length and axial stress ratio are the most influential factors contributing to the rotation and compression zone length of unbonded PT-MWs. According to the results presented in this study it is recommended to limit the axial stress ratio to a value of 0.15. In addition, limiting the maximum spacing between PT bars to a distance of six times the wall thickness is recommended to prevent local shear failure and vertical splitting cracking. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Incorporating post-tensioning into masonry offers a simple and potentially cost-effective structural system. The post-tensioning techniques can be applied to different types of masonry members as either bonded or unbonded reinforcement [1,2]. Post-tensioned (PT) walls can be ungrouted, partially grouted, or fully-grouted. In unbonded PT masonry walls, where grout is used in the cells containing the post-tensioning (PT) bars, the PT bar is enclosed in a PVC tube and hence is not embedded in or bonded to the grout. An unbonded PT bar is designed to provide a restoring force to return the wall to its original vertical alignment, therefore reducing residual drifts after a seismic event. The reduced residual drifts in unbonded post-tensioned systems result in significant reductions in repair costs and downtime, and also allow the structure to remain operational following an earthquake. The system can be used for new construction as well as repair for existing structures. For members having unbonded PT bars, the theoretical evaluation of the stress developed in the PT bars is challenging, as it ⇑ Corresponding author at: Department of Civil, Architectural & Environmental Engineering, Missouri University of Science and Technology, Rolla, MO, USA. E-mail address: [email protected] (M.A. ElGawady). http://dx.doi.org/10.1016/j.engstruct.2017.01.016 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.

depends on the elongation of the unbonded PT bar, which in turn depends on the structural element rotations. To predict the behavior of unbonded walls, empirical expressions based on finite element analyses of unbonded post-tensioned masonry walls (PTMWs) under out-of-plane and in-plane loads, were proposed [1,2]. However, the developed finite element models were not able to predict the post-peak performance of the investigated walls. Moreover, the expressions are not able to accurately predict the strength of PT-MWs having large spacing between tendons [3]. In this study, well-calibrated finite element models are used to predict the rotation of the unbonded PT-MWs at peak strength. A parametric study of theoretical walls analyzed using the finite element model was carried out in two stages. In the first stage (set I), the effects of total applied axial stress, initial stress ratio of the PT bars, height of the walls, length of the walls and spacing between PT bars were evaluated on 25 walls. During stage-two (set II), analyses of 45 additional walls were carried out to determine a relationship between the wall rotation and the compression zone length at peak strength, and other parameters including the wall configuration and the level of applied axial stress. An expression was developed based on multivariate regression analysis to predict the rotation and compression zone length at peak strength, which was integrated into a proposed equation to predict the strength of

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unbonded PT-MWs. The predicted flexural strength using the proposed design equation and the approach in which the elongation of PT bars is ignored were then compared with the values obtained from the experimental results available in the literature.

 h0 ¼

 1 f m hw 1350 f 0m Lw

ð6Þ

Using equilibrium, the compression zone length, c, can be expressed as:

P

f se i Aps i þ N abf 0m tw

2. Stress in unbonded PT bars

c¼

When an unbonded cantilever wall with a dominant flexural deformation is subjected to lateral load (Fig. 1(a)), the normal stresses on wall’s heels reduce gradually with increasing applied lateral load. If the tensile strength of masonry is ignored, the wall’s heel starts to open when the normal stress in the heel reaches zero i.e. at the decompression point. Decompression at the wall-footing interface occurs when the axial stress due to post-tensioning and gravity loads is reduced to zero at the edge of the wall base by the overturning moment resulting from the lateral loads. The PT bars display an increase in their initial post-tensioning stresses when the wall-footing interface joint opens. The elongation in PT bar i, is:

where a and b are the stress block parameters which are provided by different building codes (e.g. in MSJC 2013: a = b = 0.8). In order to determine the stress in the PT bar using Eq. (2), the values of rotation at peak strength, hm ; and compression zone length at peak strength, c, need to be determined. In this study finite element models were used to determine the relationship between c and hm and other parameters of the wall.

Di ¼ ðhm  h0 Þðdi  cÞ

ð1Þ

where hm is the wall rotation at peak strength and h0 is the rotation corresponding to the decompression point, c is the compression zone length (neutral axis depth), and di is the distance from the extreme compression fiber to the ith PT bar. Before the decompression point, the PT force remains constant, hence:

f psi ¼ f se i þ ðhm  h0 Þ

Eps ðdi  cÞ 6 f py Lps

ð2Þ

where Lps is the unbonded length of the PT bar, Eps is the Young’s modulus of the PT bar, f py is the yield stress of the PT bars and f se i is the effective stress in the ith PT bar after stress losses. (All the short term losses including anchorage slip, friction and elastic shortening, and long term losses including creep, moisture movement and steel relaxation were supposed to be considered in calculation of f se i .) Considering a linear stress-strain relationship in the masonry, assuming plane sections remain plane and ignoring the elongation of the PT bars before the decompression point, according to Fig. 1 (b), the absolute maximum masonry compressive strain corresponding to the decompression point, e0 , is:

e0 ¼

P  2 f se i Aps i þ N Lw tw Em

ð3Þ

where Aps is the area of the PT bar(s), Em is the elastic modulus of 0 0 concrete masonry = 900 f m [4] (f m : the compressive strength of masonry), tw and Lw is the thickness and length of the wall, and N is the gravity load. The axial stress, fm, is defined as:

P

fm ¼

f se i Aps i þ N Lw t w

ð4Þ 0

It has been recommended to limit f m =f m to 0.15 [5,6]. Consider0 ing a maximum value of 0.15 for f m =f m , will mean that the stress in 0 the masonry will approach the decompression point at 0.3 f m . Therefore, considering a linear stress-strain relationship in the masonry at the decompression point is a reasonable assumption. The lateral displacement at the top of the wall corresponding to the decompression state, D0 , is

D0 ¼

u0 h2w 3

ð5Þ

where u0 is the maximum value of the curvature at the decompression point = e0 =Lw , also h0 ¼ D0 =hw , hence, for concrete masonry (blocks):

ð7Þ

3. Finite element model Ryu et al. [3] developed finite element models for unbonded PTMWs. However, the model was calibrated on the seismic responses of cavity walls built using clay units. In the current work the model was calibrated and validated against single-leaf PT-MWs built using concrete masonry units (CMUs). In this study, three dimensional smeared-crack models of masonry walls were developed using LS-DYNA software, which is a general purpose finite element code. As the global behavior of the fully grouted walls was of prime importance for this study, a smeared-crack approach was applied and concrete masonry was modelled as a homogenous isotropic material using a nonlinear material model. An eight-node brick solid element was used to model the masonry components. This element includes a smeared crack for crushing in compression and cracking in tension. The material model assigned to the masonry elements1 is capable of plastic deformation, cracking in three orthogonal directions and crushing. The masonry constitutive properties are presented and discussed in the subsequent section. The concrete material for the footing and bond beam was modelled as a linear-elastic material. To simulate the PT bars, two-node beam elements with 2  2 Gauss quadrature and truss formulation were used. A bi-linear material model with linear kinematic hardening was assigned to PT bar beam elements.2 The material properties of the PT bar considered in this study were: Young’s modulus of 190 GPa, tangent modulus of 2.5 GPa, Poisson’s ratio of 0.3 and tensile yield strength of 970 MPa. Contact elements were used to model the interface between the wall and footing as well as between the wall and bond beam.3 The default contact parameters of LS-DYNA were adopted; however, as is explained below, the soft constraint option was modified to improve the reliability of the model to account for the relative stiffness of the contact elements. The assigned contact between wall and footing was capable of simulating interaction between contact interfaces of the two discrete components, which is a significant factor in capturing rocking and sliding behavior. For the stiffness of the surface to surface contact interface, a penalty-based approach was used in which the size of the contact segment and its material properties are used to determine the contact spring stiffness. This method was considered as the material stiffness parameters between the contacting surfaces were in the same range. Node-to-surface contact elements4 were also incorporated to model the interface between the PT bars and elements of the footing and bond beam. This contact element prevents the PT bars from penetrating into the solid 1 2 3 4

MAT_CONCRETE_DAMAGE_REL3. MAT_PLASTIC_KINEMATIC. AUTOMATIC_SURFACE_TO_SURFACE. AUTOMATIC_NODE_TO_SURFACE.

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Fig. 1. PT-MW (a) Before and after deformation, and (b) stress distribution at decompression of the heel.

elements. A soft constraint-based approach was used to calculate the stiffness of the node to surface interface. This approach calculates the stiffness of the linear contact springs based on the nodal masses that come into contact and the global time step size. An increasing lateral displacement5 was applied to the nodes at mid-height of the bond beam at the top of the wall. In order to simulate the connection between the footing and the laboratory strong floor, the translational and rotational degrees of freedom of the footing base nodes were fixed. 3.1. Constitutive model The basic similarities between concrete masonry and concrete tempt many researchers to use calibrated or modified concrete models to simulate masonry structures [7]. However, mortar joints represent planes of weakness in masonry structures. The effect of these planes of weakness is diminished in fully grouted walls compared to partially grouted and ungrouted walls [8]. The lack of large size coarse aggregate in fully grouted masonry is one of its main differences to concrete. The existence of large size aggregate tends to slow down the propagation of the cracks in concrete [7]. It has been reported that if an appropriate calibration is provided for masonry, a concrete model such as the one used in this study is able to accurately simulate the behavior of masonry members [3]. The K&C (Karagozian & Case) concrete-damage material model that is used in this study can accurately simulate the volumetric response of masonry material under multi-axial stress. The model is described here briefly. More detailed description of the concrete damage model can be found in [9,10]. The model decouples the deviatoric and volumetric responses. The deviatoric response includes three invariant shear failure surfaces. The current failure surface is interpolated between two of the three independent surfaces [10]. As presented in Eqs. (8)– (10), these surfaces are functions of hydrostatic pressure, P, and include the yield, maximum and residual surface, which represent the onset of damage, ultimate and residual strength of the material, respectively.

Dry ¼ a0y þ

5

P a1y þ a2y P

ðyield failure surfaceÞ

BOUNDARY_PRESCRIBED_MOTION.

ð8Þ

Drm ¼ a0 þ Drr ¼

P a1 þ a2 P

P a1f þ a2f P

ðmaximum failure surfaceÞ

ðresidual failure surfaceÞ

ð9Þ

ð10Þ

where P is equal to the mean stress defined as P ¼ 13 ðr1 þ r2 þ r3 Þ, Dri are the failure surfaces for the deviatoric stresses and are funcpffiffiffiffiffiffiffi tions of the second invariant of the deviatoric stress, J2(Dr ¼ 3J2 ), and ai values are constants defining the failure surfaces and mechanical properties of the material, which can be obtained from tests. In this study, ai-values were calibrated using test results for concrete masonry prisms as explained in the next section of the manuscript.Where g is a damage parameter which represents the extent of damage. g is defined as a function of the effective plastic strain, k. The full set of the input data of the damage function is provided in Table 1. The volumetric behavior is governed by a volumetric law through a multi-linear compaction model, which relates the hydrostatic pressure P to the volumetric strain, K. The volumetric law also prescribes a set of pressures signifying the unloading bulk modulus at peak volumetric strains. In this study, the pressures and unloading bulk moduli used for masonry were calculated using the following equation [9]:

K¼

Em 3ð1  2tÞ

ð11Þ

where t is Poisson’s ratio, and Em is the elastic modulus, taken as 0.2 0 and 900 f m [4] in this study, respectively. Table 2 presents an example of the volumetric law input values for a 13.3 MPa strong masonry. 3.2. Calibration of the material model In order to determine the parameters which characterize the masonry properties in the considered K&C material model, calibration of masonry prisms was performed versus experimental prism testing [11]. Furthermore, the model was also calibrated versus an idealized stress strain model. Priestley and Elder [12] developed a modified Kent-Park constitutive model to estimate the stressstrain relationship for confined and unconfined masonry. Their

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3.3. Validation of the finite element model (FEM)

Table 1 Input data of the damage function. k

g

0 8  106

0 0.850

24  106

0.970

40  106

0.990

56  106

1.000

72  106

0.990

88  106

0.970

550  106 0.002 0.01 0.1 1

0.005 0.005 0.005 0

0.550

Table 2 0 Volumetric law input data for f m = 13.3 MPa. Volume strain

Pressure (MPa)

0 0.0015 0.0043 0.0101 0.0305 0.0513 0.0726 0.0943 0.174 0.208

0 10 22 36 68 102 145 221 1291 1974

proposed stress-strain relationship consists of three portions described by Eqs. (12)–(14).

em < 0:0015 !

f m ðem Þ ¼ 1:067f m 0

0:0015 6 em 6 emp !

em > emp ! where :



  2em em 2  0:002 0:002

f m ðem Þ ¼ f m ½1  Z m ðem  0:0015Þ 0

f m ðem Þ ¼ 0:2 f m

Zm ¼ h

0

0:5 i

3þ0:29f 0m 145f 0m 1000

 0:002

ð12Þ

ð13Þ ð14Þ

and

emp ¼

0:8 þ em Zm

where em is the masonry compressive strain. Fig. 2(a) shows a modified Kent-Park stress-strain curve for 0 f m = 13.3 MPa and 26.5 MPa. In order to calibrate the material model, the same standard-size prism tested by Priestley and Elder [12] with the thickness/length/height values of 190 mm/390 mm/980 mm was modelled in LS-DYNA. Fig. 2(b) shows the discretization used to model the concrete masonry prism. The top and bottom nodes were fixed in the transitional x and y direction (Fig. 2(b)) to simulate the confinement induced by the testing machine platen [11]. Similar to the experiment, the top nodes were subjected to increasing axial displacement in the Z-direction (Fig. 2(b)). Fig. 2(a) shows the calibrated stressstrain relationship. As shown, the stress-strain curves obtained from the finite element analysis using the K&C material model are in good agreement with the modified Kent-Park model and experimental results. More details about the calibration can be found in Hassanli [13] and Hassanli et al. [11].

Six fully grouted unbonded PT-MWs tested by Laursen [14] were selected and analyzed to illustrate the capability and accuracy of the proposed FEM. The walls were tested as cantilever walls under static cyclic load. The dimensions and configurations of the considered walls are presented in Table 3. During the finite element modeling, the lateral load was applied to the mid-height of the cap beam of each wall using displacement control. The applied lateral displacement was increased until failure occurred. Explicit static analysis was used and sensitivity analyses were conducted to determine the optimum applied displacement rate to avoid inertial effects. Densities of 2000 kg/ m3 and 2400 kg/m3 were considered in the material model of masonry and concrete, respectively. According to the material testing the PT bar had density, yield strength, tensile strength, elastic modulus and tangent modulus of 8000 kg/m3, 970 MPa, 1160 MPa and 190 GPa and 2.5 GPa, respectively, which were implemented in the model. Depending on the wall dimensions, the FEM of the walls consisted of 3767–5375 elements and 5547–7717 nodes and the average element size was 80 mm. Sensitivity analysis was carried out to determine the element size, until further reducing the element size had no effect on the global response of the walls. The lateral force-displacement obtained from the FEMs and the backbone curves obtained from the experimental cyclic forcedisplacement response for pull and push directions are plotted in Fig. 3. As shown in the figure, the model can predict the wall strength, initial stiffness and rotational capacity with good accuracy. The lateral strength of the wall obtained during testing, VEXP, and obtained from the FEM, VFEM, are presented in Table 3. As shown in the table VEXP/VFEM ranged from 0.85 to 1.04. Moreover, as presented in Table 3, comparing the experimental results and the finite element model results indicates that the total PT forces predicted by the FEMs fell within 10% of the experimental results. The FEM reasonably captured the damage pattern of the PTMWs. According to the experimental work, a flexural failure mode and gradual strength degradation was observed for all walls (except for wall W6), which was attributed to the spalling of the face shell and crushing of the grout core at the toe zone. The same failure mode was observed in the FEM. Fig. 4 presents a schematic illustration of damage at the failure stage observed in the experiment and compared with the extent of damage obtained from the finite element analysis. Both the test results and FEM showed that the plastic deformation was confined to the lowest two masonry courses, equivalent to 400 mm height, except for wall W6. Of all the walls, only wall W1 included shear reinforcement at the mid-height. Both the experimental results and FEM showed that no shear cracks developed in the wall and hence the shear reinforcement was not engaged at all (Fig. 4(a) and (b)). Some minor cracks formed at the position of the side PT bars in the tests were not captured in the FEM, which can be attributed to the homogenization of the material used in the FEM. These cracks provided some local effects but they did not influence the global response. Both the experimental results and FEM showed a splitting crack in the case of wall W2 which was attributed to large localized splitting forces of the pre-stress anchorage. The model showed a splitting tensile failure along the side PT bars, while it occurred along the central PT bar in the test. Of all walls, only wall W6 exhibited a brittle failure mode during the experimental work. The failure mode was characterized by diagonal cracking due to tensile splitting of the masonry compression struts between the post-tensioning anchorage at the top and the toe of the wall. However, wall W6 provided a rocking mechanism and flexural behavior before it reached a drift of 0.5%. Fig. 5 compares the maximum inplane stress contour obtained for wall W5 and wall W6 at the peak

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Axial stress (MPa)

30 FEM Test 1 Test 2 Test 3 Modified Kent-Park

25 20 15 10 5 0

f'm=26.5 MPa 0

0.2

0.4

0.6

0.8

1

Strain (%) 14 FEM Modified Kent-Park

Stress (MPa)

12 10 8 6 4 f'm=13.3 MPa

2

(b)

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Strain (%)

(a) Fig. 2. Prism model (a) Stress-strain behavior of prism under axial compression load, and (b) FE mesh model.

Table 3 Configuration of the selected experimental walls. Wall

tw (m)

Lw (m)

f0 m (MPa)

No. of PT Bars

PT bar initial stress (fse) (MPa)

Initial PT force on the wall (kN)

Axial stress ratio (fm/f0 m)

VEXP (kN) Push

Pull

Average

W1 W2 W3 W4 W5 W6

0.19 0.14 0.14 0.14 0.14 0.14

3 3 3 3 1.8 1.8

13.3 15.1 20.6 17.8 20.5 18.4

3 3 2 2 2 3

468 555 757 757 534 614

580 690 622 628 445 760

0.077 0.109 0.072 0.084 0.086 0.164

550 472 378 386 183 267

536 486 368 390 193 249

543 479 373 388 188 258

VFEM (kN)

VEXP/VFEM

Total PT force at peak strength (kN) TEXP

TFEM

TEXP/TFEM

547 460 427 415 221 268

0.99 1.04 0.87 0.94 0.85 0.96

842 776 697 658 592 870

847 795 689 702 657 906

0.99 0.98 1.01 0.94 0.90 0.96

The typical height of the tested specimens was 2800 mm. The yield and ultimate strength were 970 MPa and 1160 MPa, respectively.

strength. Fig. 4 shows the damage pattern and plastic contours at the rupture of the walls while Fig. 5 shows the stress contours at the peak load. Both wall W5 and wall W6 had the same configuration, but were subjected to different levels of axial stress ratio due to different numbers of PT bars. The stress contours indicate a direct compression strut from the pre-stressing anchorage and toe of wall W6, which did not appear in wall W5. Moreover, according to the force-displacement FEM results, of all considered walls, wall W6 was the only wall which exhibited no yielding of the tensile PT bars, which is consistent with the experimental results. For the tested walls, although similar strengths were recorded in pull and push directions during the experimental work, there was a significant difference in displacements corresponding to the peak strength in pull and push directions. For wall W1, for example, while in the pull direction the displacement at peak strength was 24.07 mm, it was only 12.26 mm in the push direction. Hence, it is very challenging when it comes to validate the FEM results. For example, the displacement at peak strength

obtained from the FEM for Wall W1 was 15.46 mm, which is located between the two experimental values. Similarly, although the FEM model could not capture the rotation at peak strength of walls W2 and W3 accurately, the rotation prediction falls within the range of rotations in the experiment corresponding to strength higher than 90% of the wall strength. Hence, the error in the rotation prediction at peak strength obtained from the FEM is acceptable, as in this study the rotation values at peak strength are only used to predict the wall strength. 4. Parametric study The validated finite element model was then used to conduct a parametric study to develop expressions for the rotation and flexural strength of PT-MWs. The parametric study was performed in two stages by considering two different sets of wall models. The first set was developed to evaluate the effect of different parameters on the flexural strength and drift capacity of walls at peak strength. Based on the analysis results obtained from this set,

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R. Hassanli et al. / Engineering Structures 136 (2017) 245–260

600

(a)

500

Wall w1

Shear Force (kN)

Shear Force (kN)

600

400 300 200 FEM Exp-pull Exp-push

100 0 0

10

20

30

40

500

(b)

400 300 200 FEM Exp-Pull Exp-Push

100 0

50

0

10

Lateral Displacment (mm) 600

(c)

500

Wall w3

400 300 200 FEM Exp-pull Exp-push

100 0 0

10

20

30

300 200 FEM Exp-pull Exp-push

100

50

0

10

20

30

Wall w5

200 100 0

(f)

500

10

20

30

40

50

Wall w6

400 300 200

FEM Exp-pull Exp-push

100 0

0

40

Lateral Displacment (mm)

Shear Force (kN)

Shear Force (kN)

300

50

Wall w4

0

40

FEM Exp-pull Exp-push

400

40

400

600

(e)

500

30

(d)

500

Lateral Displacment (mm) 600

20

Lateral Displacment (mm)

Shear Force (kN)

Shear Force (kN)

600

Wall w2

50

Lateral Displacment (mm)

0

10

20

30

40

50

Lateral Displacment (mm)

Fig. 3. Experimental and calculated force displacement response (a) wall W1, (b) wall W2, (c) wall W3, (d) wall W4, (e) wall W5, and (f) wall W6.

design recommendations were provided for PT-MWs. Subsequently, matrices of the wall models were defined for the second set of the parametric study. Following this step, expressions for rotation and a design method to evaluate the flexural strength of unbonded PT-MWs were developed. 4.1. Parametric study-set I The first stage of the parametric study included 25 walls in five groups (Table 4) to assess the influence of different parameters on strength and deformation. As presented in Table 4, the walls W1-1 to W1-5, W2-1 to W2-8, W3-1 to W3-4, W4-1 to W4-5 and W5-1 0 to W5-3 were used to study the effects of axial stress ratio, f m =f m , initial to yield stress ratio of PT bars, fi/fpy, height of the wall, hw, spacing of the PT bars and the length of the wall, respectively, on the strength and deformation of the wall. A wall with the dimensions height/thickness/length of 2800 mm/190 mm/3000 mm was considered as the ‘‘control” specimen. The nominal compressive strength of the masonry and the thickness of the wall in both sets of the parametric study were taken as 13.3 MPa and 190 mm, respectively. 0

4.1.1. Effect of axial stress ratio, f m =f m Five different values of axial force of 204.8 kN, 577.5 kN, 831.0 kN, 1060.5 kN and 2112.0 kN corresponding to 2.7%, 7.6%, 0 11.0%, 14.0% and 27.9% of f m were considered for walls W1-1 to W1-5, respectively. Fig. 6(a) presents the base shear versus displacement obtained for these walls from the FEM. As shown in

0

the figure, as the f m =f m ratio increases, the strength increases while 0 the ductility decreases. Walls with higher ratios of f m =f m experienced a sudden drop in their lateral resistances beyond the peak strength. Fig. 6(b) illustrates the effect of the axial stress ratio on the lateral strength of PT-MWs. As shown in the figure, the strength increase due to an increase in the axial stress ratio is more 0 pronounced until f m =f m reaches a value of 0.15. Beyond this value, the axial stress ratio has a relatively small effect on the in-plane strength. Hassanli et al. [5] developed similar conclusions based on analysis of experimental results of 31 walls specimens. More0 over, a higher value of f m =f m results in a lesser ductility value. Fig. 7(a) and (b) presents the stress contours at the peak strength of the walls W1-4 and W1-5. While for the wall W1-4 having an axial stress ratio of 0.14, a stress concentration at the toe indicates a flexural mode of failure, for the wall W1-5, having an axial stress ratio of 0.28, stress concentrations at the toe and middle of the wall indicate a brittle splitting mode of failure. This is consistent with the results reported by Laursen [14] and Hassanli et al. [5], that a brittle failure might occur by increasing the axial stress ratio. The increased axial load causes diagonal cracking due to tensile splitting of the masonry compression strut that forms between the posttensioning anchorage of the loading beam and the wall flexural compression zone. 4.1.2. Effect of the initial to yield stress ratio of PT bars f i =f py To investigate the effect of the initial to yield stress ratio in the PT bars, walls W2-1 to W2-8 were generated. For these walls all parameters including the number of PT bars and the wall configuration

R. Hassanli et al. / Engineering Structures 136 (2017) 245–260

(a) Wall w1 – Experiment (Laursen 2002)

251

(b) Wall w1 – FEM

Potential vertical splitting crack

(c) Wall w2 – Experiment (Laursen 2002)

(d) Wall w2 – FEM

(e) Wall w3 – Experiment (Laursen 2002)

(f) Wall w3 - FEM

Fig. 4. Comparison of damage patterns observed in experiment and FEM.

were made the same, and the only variable was fi/fpy. Four different values of fi/fpy = 0.29, 0.48, 0.66, 0.83 were considered. To provide the same axial stress ratio for walls W2-1 to W2-4, the total initial PT force was kept the same; however the yield strength of the PT bars was varied, as presented in Table 4. To provide the same axial stress ratio for walls W2-5 to W2-8, the total initial PT force was kept the same by using different cross-sectional areas of PT bars; however, the initial stress of the PT bars was varied. A higher level of fi/fpy resulted in a smaller strength and a greater ductility compared to walls having smaller values of fi/fpy (Fig. 8). The PT bar in walls having high values of fi/fpy yielded at a small drift value, while in the case of small values of fi/fpy, higher values of PT force were developed prior to yielding, resulting in a higher lateral strength of the wall. Hence, the fi/fpy ratio was limited to 0.6 in generating the wall configuration for wall set II of the parametric study. It is worth noting that it is common practice in the U.S. to limit the post-tensioning force, after initial losses, to 45–65% of fpy. Hence, this recommendation is in line with current practice. 4.1.3. Effect of the height, hw To investigate the effect of the height on the behavior of PTMWs, walls W3-1 to W3-4 having heights ranging from

1800 mm to 4800 mm were investigated (Table 4). Fig. 9(a) presents the base shear versus displacement results for these walls. As shown in the figure, while by increasing the height of the wall the displacement capacity increases, the base shear capacity decreases. To normalize the response, the base moment versus drift curve is presented in Fig. 9(b). As shown in the figure, the peak moment strength and the drift corresponding to the peak strength is approximately the same for all walls, regardless of the height of the wall. Consequently, within the range of the parameters considered in this study, it seems that the drift capacity of the wall at peak strength is not a function of the height of the wall. Fig. 10 presents the base crack profile of the walls W3-1 to W34 at the peak strength. As shown in the figure, the compression zone length is approximately the same for all considered walls. The compression zone length, c, for walls W3-1, W3-2, W3-3 and W3-4 was found to be equal to 349.7 mm, 353.0 mm, 354.4 mm and 363.2 mm, corresponding to 11.66%, 11.77%, 11.81% and 12.10% of Lw, respectively. The variation is within two percent of the average value of c and can therefore be ignored. Hence, the compression zone length, c, can be considered to be independent of the height of the wall.

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R. Hassanli et al. / Engineering Structures 136 (2017) 245–260

(g) Wall w4 – Experiment (Laursen 2002)

(h) Wall w4 – FEM

(i) Wall w5 – Experiment (Laursen 2002)

(j) Wall w5 - FEM

(k) Wall w6 – Experiment (Laursen 2002)

(l) Wall w6 – FEM

Fig. 4 (continued)

Direct strut (a) Wall w5-Flexural failure

(b) Wall w6-Compression strut failure

Fig. 5. Comparison of maximum in-plane stress contours.

4.1.4. Effect of the PT bar spacing As presented in Table 4, walls W4-1 to W4-5 were generated to investigate the influence of the spacing between PT bars. As shown

in Table 4, the number of PT bars considered in walls W4-1 to W42 were, two, three, four, five and eight, corresponding to spacing of 4400 mm, 2200 mm, 1400 mm, 1100 mm, and 600 mm,

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R. Hassanli et al. / Engineering Structures 136 (2017) 245–260 Table 4 Wall matrix of set I-parametric study.a Wall

h (mm)

tw (mm)

Lw (mm)

fm/f0 m

No. of PT bars

fpy (MPa)

fpu (MPa)

fse (MPa)

Axial force (kN)

fi/fy

0

fm/f m

W1-1 W1-2 W1-3 W1-4 W1-5

2800 2800 2800 2800 2800

190 190 190 190 190

3000 3000 3000 3000 3000

0.027 0.076 0.110 0.140 0.279

3 3 3 3 3

970 970 970 970 970

1160 1160 1160 1160 1160

495 466 447 428 352

204.8 577.5 831.0 1060.5 2112.0

0.51 0.48 0.46 0.44 0.36

fi/fyb

W2-1 W2-2 W2-3 W2-4

2800 2800 2800 2800

190 190 190 190

3000 3000 3000 3000

0.076 0.076 0.076 0.075

3 3 3 3

1595 970 709 555

1907 1160 848 663

466 466 466 459

577.5 577.5 577.5 568.5

0.29 0.48 0.66 0.83

fi/fyc

W2-5 W2-6 W2-7 W2-8

2800 2800 2800 2800

190 190 190 190

3000 3000 3000 3000

0.076 0.076 0.076 0.075

3 3 3 3

970 970 970 970

1160 1160 1160 1160

466 466 466 459

577.5 577.5 577.5 568.5

0.29 0.48 0.66 0.83

hw

W3-1 W3-2 W3-3 W3-4

1800 2800 3800 4800

190 190 190 190

3000 3000 3000 3000

0.074 0.076 0.077 0.078

3 3 3 3

970 970 970 970

1160 1160 1160 1160

454 466 472 478

562.5 577.5 585.0 592.5

0.47 0.48 0.49 0.49

PT bar spacing

W4-1 W4-2 W4-3 W4-4 W4-5

2800 2800 2800 2800 2800

190 190 190 190 190

5000 5000 5000 5000 5000

0.068 0.073 0.074 0.052 0.069

2 3 4 5 8

970 970 970 970 970

1160 1160 1160 1160 1160

414 446 451 476 477

856.0 921.0 932.0 655.0 876.0

0.43 0.46 0.47 0.49 0.49

Length

W5-1 W5-2 W5-3

2800 2800 2800

190 190 190

1800 3000 4200

0.077 0.076 0.078

3 3 3

970 970 970

1160 1160 1160

468 466 456

348.0 577.5 824.0

0.48 0.48 0.47

Variable

a b c

Typical unbonded length = hw + 800 mm, Typical PT bar diameter ranges from to 13 mm to 50 mm. fy variable, fi constant. fi variable, fy constant.

Dri % 1000

1

900

fm/f'm=0.278

800

Base shear (kN)

2

fm/f'm=0.351 fm/f'm=0.140

700

fm/f'm=0.110

600 500

fm/f'm=0.076

400 300

fm/f'm=0.027

200

(b)

1000

Base shear (kN)

(a)

0

800 600 400 200 0 0

100

0.1

0.2

0.3

0.4

fm/f'm

0 0

20

40

60

Displacement (mm) Fig. 6. Effect of axial stress ratio on (a) force-displacement response, and (b) lateral strength.

respectively. Fig. 11 compares the damage pattern of the walls at the maximum strength. Comparing the damage pattern reveals that walls with PT bar spacing of more than 1400 mm, corresponding to 7tw, would suffer vertical splitting cracks in the compression zone near the location of the farthest PT bar. A flexural failure mode is exhibited by wall W4-2 in which the PT bars are spaced more closely. Hence, the PT bar spacing should be limited to a value between 1100 mm and 1400 mm. This is in line with the recommendations of MSJC 2013 [4] for PT walls designed for out-ofplane loads. Hence, for set II of parametric study, the maximum spacing between PT bars was considered to be equal to 6tw. 4.1.5. Effect of wall length, Lw Walls W5-1 to W5-3 with lengths of 1800 mm, 3000 mm and 4200 mm respectively, were considered to investigate the effect of the wall length on the drift and flexural strength (Table 4).

Fig. 12(a) compares the force vs displacement curves of the walls with different lengths, and Fig. 12(b) presents the base shear nor0 malized by f m An versus drift; where An is the net cross sectional area of the wall. As shown in the figure, by increasing the wall length while the strength increases, the ductility reduces. Consequently, the wall length has an influential effect on the drift capacity, strength and general behavior of the PT-MWs.

4.2. Parametric study-set II The configuration of the walls in set II of the parametric study (Table 5) was determined according to the conclusions obtained from set I of the parametric study. As shown in Table 5, three different lengths of 1800 mm, 3000 mm and 4200 mm and three different heights of 2800 mm, 3800 mm and 4800 mm were

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R. Hassanli et al. / Engineering Structures 136 (2017) 245–260

Fig. 7. Damage pattern (a) wall W1-4, (b) wall W1-5.

Dri %

Dri % 0.5

1

0

2

(b)

fi/fpy=0.29

600

Base shear (kN)

1.5

fi/fpy=0.48

500

fi/fpy=0.66

400 300 200 100 0

20

40

1

700

1.5

2

fi/fpy=0.29 fi/fpy=0.48

500

fi/fpy=0.66

400 300 200 100

fi/fpy=0.83

0

0.5

600

Base shear (kN)

(a) 700

0

fi/fpy=0.8

0 0

60

Displacement (mm)

20

40

60

Displacement (mm)

Fig. 8. Effect of initial to yield stress ratio in PT bars (a) fi constant, fpy variable and (b) fi variable, fpy constant.

900 800 700 600 500 400 300 200 100 0

(b) 1600

W3-1 W3-2 W3-3 W3-4

Base moment (kN.m)

Base shear (kN)

(a)

1400 1200 1000 800 600

W3-1 W3-2 W3-3 W3-4

400 200 0

0

20

40

60

0

0.5

1

1.5

Dri %

Displacement (mm)

Fig. 9. (a) Force-displacement curve and (b) moment-drift curve, for walls of set I having different heights.

considered in this set for analysis, the axial stress ratio was varied from 0.05% to 0.15%, and the fi/fpy value ranged from 0.3 to 0.6. 4.3. Proposed expression to predict the flexural strength of unbonded masonry walls The wall’s rotation at peak strength, hm, and the compression zone length at peak strength, c, obtained from the finite element analysis are presented in Table 5. According to the results, hm, ranged from 0.46% to 1.53%, and the compression zone length, c, ranged from 160 mm to 1375 mm, corresponding to 9.7% and 32.7% of the wall length, Lw, respectively. Different factors contribute to the rotation and the compression zone length at peak strength including: length, height and thick-

ness of the wall, axial stress ratio, longitudinal and shear reinforcement ratio, steel, concrete/masonry material properties, level of confinement, loading pattern and moment gradient. Considering the walls failed in flexure and using the values obtained for hm and c for the different wall configurations (Table 5), a multivariate regression analysis was carried out. As a result, the wall’s length and the axial pre-stress ratio were found to be the most influential factors affecting hm and c. Considering these two parameters, the following equation was obtained from the multivariate regression analysis to estimate the rotation of unbonded PT-MWs at their peak strength:

hm c ¼

f 0:00055Lw þ 17:375 m 0 fm

!

ð15Þ

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R. Hassanli et al. / Engineering Structures 136 (2017) 245–260

2

Base displacement (mm)

20.0

f psi ¼ f sei þ 4

W3-1 W3-2 W3-3 W3-4

15.0 10.0

0:00055Lw þ 17:375 ff 0m m

c

 h0 5

Eps ðdi  cÞ 6 f py Lps

ð16Þ

Using Eq. (6), the values of h0 for the walls of set II of the parametric study were obtained and presented in Table 5. As presented in the table, h0 is considerably smaller than hm, and hence can be ignored in comparison with hm. Therefore, the following Equation is proposed to predict the stress developed in an unbonded PTMW at peak strength:

5.0 0.0

f psi ¼ f sei þ 0:00055Lw þ 17:375

-5.0 0

3

1000

2000

3000

Wall length (mm) Fig. 10. Base crack profile.

The obtained values for R2 (coefficient of determination) and adjusted-R2 (adjusted coefficient of determination) were both higher than 90%. Design of experiments (DOE) data showed a low interaction between the parameters. The application of this expression is limited to wall members having a flexural mode of failure. Fig. 14(a) and (b) indicates the effect of the wall length and axial stress ratio on the hm c of set II of walls. The upward trend of data presented in Fig. 13(a) reveals that as the wall length increases the hm c value increases. Moreover, according to Fig. 13(b) and Eq. (15), the hm c value for members having a higher level of axial stress ratio, is larger. According to Fig. 1(a), considering a constant value for masonry ultimate strain, emu , the plastic hinge length, Lpl, is proportional to hm c. (hm c ¼ emu Lpl ). Combining Eq. (15) with Eq. (2) gives,

!   f m Eps di  1 6 f py 0 f m Lps c

The following steps can be considered to estimate the strength of a PT-MW: Assume f ps ¼ f se , Determine the compression length, c, using Eq. (7), Determine the stress in the tendons, f ps i , using Eq. (17), Perform an iteration of steps 2 and 3 to reach a constant value of c, (5) Determine the moment capacity, Mn , using Eq. (18) and using the c and f ps i values obtained in step 4.

(1) (2) (3) (4)

Mn ¼

X

f se Aps ¼

X

   a Lw a þN f ps i Aps i di   2 2 2

(a) Wall W4-1 (spacing=4600mm)

(b) W4-2 (spacing=2200mm)

Potential vertical splitting crack

(c) Wall W4-3 (spacing=1400mm)

ð18Þ

where a ¼ bc, in which b is stress block parameter. The method can be easily applied to design software as there is an iterative approach involved in the design procedure. The iteration process is required as c and f ps i are inter-related parameters in Eqs. (7) and (17). Note that the stresses in the PT bars located in the compression zone are less than the initial stresses.

Potential vertical splitting crack

Potential vertical splitting crack

ð17Þ

(d) Wall W4-4 (spacing=1100mm)

Fig. 11. The effect of spacing on failure mode and damage pattern.

R. Hassanli et al. / Engineering Structures 136 (2017) 245–260

Base shear (kN)

1200

(a)

1000

Normalized base shear

256

W5-1 W5-2 W5-3

800 600 400 200 0 0

20

40

1.6

(b)

W5-1 W5-2 W5-3

1.2 0.8 0.4

60

0 0

0.5

Displacement (mm)

1

1.5

2

Dri %

Fig. 12. (a) Force-displacement curve and (b) normalized based shear versus drift of walls of set I, having different lengths.

Table 5 Wall matrix of set II-parametric study. Wall name

hw (m)

Lw (m)

Self-weight (kN)

fm/f0 m

No. of bars

Lps (mm)

Initial stress, fse (MPa)

Total pre-stressing force (kN)

h m%

c (m)

h0%

W1.8-2.8-1 W1.8-2.8-2 W1.8-2.8-3 W1.8-2.8-4 W1.8-2.8-5 W1.8-3.8-1 W1.8-3.8-2 W1.8-3.8-3 W1.8-3.8-4 W1.8-3.8-5 W1.8-4.8-1 W1.8-4.8-2 W1.8-4.8-3 W1.8-4.8-4 W1.8-4.8-5 W3.0-2.8-1 W3.0-2.8-2 W3.0-2.8-3 W3.0-2.8-4 W3.0-2.8-5 W3.0-3.8-1 W3.0-3.8-2 W3.0-3.8-3 W3.0-3.8-4 W3.0-3.8-5 W3.0-4.8-1 W3.0-4.8-2 W3.0-4.8-3 W3.0-4.8-4 W3.0-4.8-5 W4.2-2.8-1 W4.2-2.8-2 W4.2-2.8-3 W4.2-2.8-4 W4.2-2.8-5 W4.2-3.8-1 W4.2-3.8-2 W4.2-3.8-3 W4.2-3.8-4 W4.2-3.8-5 W4.2-4.8-1 W4.2-4.8-2 W4.2-4.8-3 W4.2-4.8-4 W4.2-4.8-5

2.8 2.8 2.8 2.8 2.8 3.8 3.8 3.8 3.8 3.8 4.8 4.8 4.8 4.8 4.8 2.8 2.8 2.8 2.8 2.8 3.8 3.8 3.8 3.8 3.8 4.8 4.8 4.8 4.8 4.8 2.8 2.8 2.8 2.8 2.8 3.8 3.8 3.8 3.8 3.8 4.8 4.8 4.8 4.8 4.8

1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2

23 23 23 23 23 31 31 31 31 31 39 39 39 39 39 38 38 38 38 38 52 52 52 52 52 66 66 66 66 66 54 54 54 54 54 73 73 73 73 73 92 92 92 92 92

0.052 0.077 0.098 0.12 0.14 0.052 0.078 0.1 0.122 0.144 0.053 0.078 0.101 0.124 0.146 0.052 0.076 0.097 0.119 0.139 0.052 0.077 0.099 0.121 0.142 0.053 0.078 0.101 0.123 0.145 0.054 0.078 0.096 0.121 0.139 0.054 0.079 0.099 0.124 0.145 0.055 0.08 0.101 0.127 0.149

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 2.6 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

482.3 467.9 455.8 444.7 433.3 486.4 476.0 463.9 454.6 445.5 489.5 478.7 470.1 461.1 452.3 533.4 517.6 502.8 490.5 476.9 539.5 524.9 513.0 500.3 489.1 543.6 530.3 519.5 508.8 498.7 582.4 553.9 523.2 523.2 504.1 588.6 564.7 537.5 539.6 525.9 596.8 572.8 545.7 552.7 539.5

236 348 446 544 636 238 354 454 556 654 239.5 356 460 564 664 391.5 577.5 738 900 1050 396 585.6 753 918 1077 399 591.6 762.6 933.6 1098 570 824 1024 1280 1480 576 840 1052 1320 1544 584 852 1068 1352 1584

0.86 0.92 0.89 0.89 0.93 1.22 1.15 1.02 1.02 0.87 1.24 1.16 1.03 1.03 0.83 1.07 0.68 0.58 0.58 0.55 1.08 0.64 0.54 0.54 0.53 0.9 0.64 0.56 0.56 0.51 0.72 0.59 0.47 0.47 0.46 1.43 1.12 0.93 0.93 0.85 1.53 1.17 0.89 0.89 0.84

0.18 0.18 0.26 0.38 0.39 0.16 0.22 0.32 0.39 0.37 0.16 0.26 0.31 0.39 0.36 0.28 0.40 0.59 0.59 0.68 0.29 0.36 0.50 0.59 0.69 0.27 0.40 0.49 0.59 0.68 0.47 0.79 0.89 0.95 1.38 0.46 0.59 0.78 0.88 0.97 0.39 0.62 0.77 0.89 0.96

6.E03 9.E03 1.E02 1.E02 2.E02 8.E03 1.E02 2.E02 2.E02 2.E02 1.E02 2.E02 2.E02 2.E02 3.E02 4.E03 5.E03 7.E03 8.E03 1.E02 5.E03 7.E03 9.E03 1.E02 1.E02 6.E03 9.E03 1.E02 1.E02 2.E02 3.E03 4.E03 5.E03 6.E03 7.E03 4.E03 5.E03 7.E03 8.E03 1.E02 5.E03 7.E03 9.E03 1.E02 1.E02

Typical unbonded length = hw + 800 mm. Typical PT bar diameter ranges from to 25 mm to 61.16 mm.

Note that Eq. (17) was developed for purely unbonded PT-MWs. For hybrid walls, the effect of conventional reinforcement also needs to be considered.

According to MSJC 2013, the base shear capacity of a PT-MW that does not incorporate bonded reinforcement, is the minimum strength obtained from the shear expression (Eq. (19)) and the flexural expression.

4.4. Validation of the proposed design approach The strength predicted by the proposed design approach has been validated against experimental results as well as finite element model results and compared with the prediction of the MSJC [4].

V n ¼ min

8 qffiffiffiffiffiffi 0 > > < 0:315An f m 2:07An > > : 0:621An þ 0:45N

ðaÞ ðbÞ ðcÞ

ð19Þ

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R. Hassanli et al. / Engineering Structures 136 (2017) 245–260

(a)

2000

(b)

1500

θmc (mm)

θmc (mm)

1500

2000

1000 500

1000 500

0

0 0

2000

4000

6000

0

0.05

L w (mm)

0.1

0.15

0.2

f'm/fm

Fig. 13. Effect of the (a) length and (b) axial stress ratio on the plastic hinge length.

Table 6 Post-tensioned masonry wall database. Ref.

Wall designation

Original designation

Specification

Material unit

Loading

Laursen [14]

L1-Wall1 L1-Wall2 L1-Wall3 L1-Wall4 L1-Wall5 L1-Wall6 L3- Wall1 L3-Wall2 L2-Wall1 L2-Wall2 L2-Wall5

FG:L3.0-W20-P3 FG:L3.0-W15-P3 FG:L3.0-W15-P2C FG:L3.0-W15-P2E FG:L1.8-W15-P2 FG:L1.8-W15-P3 S3-1 S3-2 FG:L3.0-W15-P1-CP FG:L3.0-W15-P2-CP FG:L3.0-W15-P2-HB

FG FG FG FG FG FG FG + Confinement FG + Confinement FG + Confinement FG + Confinement FG + Confinement

CMU CMU CMU CMU CMU CMU CMU CMU CMU CMU CMU

Cyclic Cyclic Cyclic Cyclic Cyclic Cyclic Cyclic Cyclic Cyclic Cyclic Cyclic

R-Wall1 R- Wall2 R- Wall3

Test1 Test3 Test2

FG FG + Confinement plate FG + supplemental mild steel

Brick Brick Brick

Cyclic Cyclic Cyclic

Rosenboom [16]

plate plate plate plate plate + high strength block

FG = fully grouted. CMU = concrete masonry unit.

600

1.20

400

VEQN/VEXP

VEQN (kN)

(a)

200

(b)

1.00 0.80 0.60

Proposed approach MSJC2013

0 0

200

400

Proposed approach MSJC2013

0.40 600

0

0.1

0.2

0.3

Axial stress rao

VEXP (kN)

Fig. 14. (a) Comparison of the strength prediction and (b) comparison of VEQN/VEXP, using the proposed approach and MSJC2013 approach based on experimental results.

The nominal moment capacity of a PT-MW using MSJC 2013 is:

Mn ¼

X

   a Lw a þN f se Aps d   2 2 2

ð20Þ

P where :

a¼

f se Aps þ N 0 0:8tw f m

The predicted lateral strength of PT-MWs using the flexural expression is equal to the moment capacity, Mn, divided by the effective height, hw. As shown in Eq. (20), the MSJC 2013 does not take into account the distribution of PT bars or the stress increment due to the elongation of PT bars. However, the proposed expression (Eq. (18)) considers both the elongation of the PT bars and their distribution along the length of the wall.

4.4.1. Validation of the proposed design approach against experimental results To verify the accuracy of the proposed method compared with experimental results, following a comprehensive literature review of PT-MWs, a database of 14 unbonded post-tensioned fully grouted specimens was collected. The summary of the selected walls is presented in Table 6. Fig. 14(a) and (b) compares the VEQN/VEXP calculated using the proposed method and the MSJC method versus axial stress ratio. VEQN is the minimum of the strength obtained from the shear equation (Eq. (19)) and flexural expression. As shown, as the axial stress ratio increases, the prediction, without taking into account the elongation of PT bars, becomes more conservative. However, the prediction from the proposed method is unbiased toward the axial stress ratio and hence the proposed equation has effectively improved the strength prediction. While the value

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R. Hassanli et al. / Engineering Structures 136 (2017) 245–260

4.4.2. Validation of the proposed design approach against finite element results The FEM results of walls set I and set II, were also used to investigate the accuracy of ignoring the elongation of PT bars in MSJC 2013 and compare it with the results obtained using the proposed approach. The base shear determined using the two wall sets are presented in Tables 7 and 8 and Fig. 15. The mode of failure and the ratio of VEQN/VFEM is also provided in Tables 7 and 8.

of VEQN/VEXP considering no elongations of PT bars (MSJC 2013) varies from 0.54 to 1.05 with an average value of 0.75 and a coefficient of variation (COV) of 17.0%, using the proposed approach it varies from 0.86 to 1.09 with an average of 0.96 and a COV of 8.6%. The predicted strength obtained from the proposed equation falls within ±15% of the test results. Moreover, the proposed approach has reduced the scatter of the data compared with MSJC 2013.

Table 7 Strength prediction-Set I of PT-MWs. Wall name

VFEM (kN)

Failure mode

Proposed approach

MSJC 2013 Flexural

W1-1 W1-2 W1-3a W1-4a W1-5a W2-1 W2-2 W2-3 W2-4 W2-5a W2-6 W2-7 W2-8 W3-1a W3-2 W3-3 W3-4 W4-1a W4-2a W4-3a W4-4a W4-5a W5-1 W5-2 W5-3a a

200 560 714 823 867 601 560 451 348 644 560 400 330 797 560 386 300 989 1130 1200 955 1090 193 560 934

Flexure Flexure Flexure Shear Shear Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Shear Flexure Flexure Flexure Shear Shear Shear Shear Shear Flexure Flexure Shear

Shear

VEQN (kN)

VEQN/VFEM

VEQN (kN)

VEQN/VFEM

VEQN (kN)

VEQN/VFEM

198 500 665 778 1202 526 503 407 327 622 500 405 330 799 500 361 271 1573 1538 1476 1053 1430 145 500 1076

0.99 0.89 0.93 0.95 1.39 0.87 0.90 0.90 0.94 0.97 0.89 1.01 1.00 1.00 0.89 0.93 0.90 1.59 1.36 1.23 1.10 1.31 0.75 0.89 1.15

106 280 384 469 737 280 280 280 276 280 280 280 280 425 280 209 167 699 748 755 547 714 88 280 543

0.53 0.5 0.54 0.57 0.85 0.47 0.5 0.62 0.79 0.43 0.5 0.7 0.85 0.53 0.5 0.54 0.56 0.71 0.66 0.63 0.57 0.66 0.46 0.5 0.58

446 614 655 655 655 614 614 614 610 614 614 614 610 607 614 617 621 975 1005 1009 885 984 369 614 852

2.23 1.10 0.92 0.80 0.76 1.02 1.10 1.36 1.75 0.95 1.10 1.53 1.86 0.76 1.10 1.60 2.07 0.99 0.89 0.84 0.93 0.90 1.91 1.10 0.91

Lateral strength using proposed flexural expression is less than the shear expression of MSJC (2013).

Table 8 Strength prediction-Set II of PT-MWs. Wall name

VFEM (kN)

Failure mode

Proposed approach

MSJC 2013 Flexural

W1.8-2.8-1 W1.8-2.8-2 W1.8-2.8-3 W1.8-2.8-4 W1.8-2.8-5 W1.8-3.8-1 W1.8-3.8-2 W1.8-3.8-3 W1.8-3.8-4 W1.8-3.8-5 W1.8-4.8-1 W1.8-4.8-2 W1.8-4.8-3 W1.8-4.8-4 W1.8-4.8-5 W3.0-2.8-1 W3.0-2.8-2 W3.0-2.8-3 W3.0-2.8-4 W3.0-2.8-5a W3.0-3.8-1 W3.0-3.8-2

138 200 257 297 331 100 166 186 200 215 78.8 126 145 149 164 389 538 647 727 793 293 400

Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Shear Flexure Flexure

Shear

VEQN (kN)

VEQN/VFEM

VEQN (kN)

VEQN/VFEM

VEQN (kN)

VEQN/VFEM

135 179 212 243 269 95 128 148 170 188 72 95 113 130 144 348 479 588 683 753 255 351

0.98 0.90 0.83 0.82 0.81 0.95 0.77 0.80 0.85 0.88 0.92 0.75 0.78 0.87 0.88 0.89 0.89 0.91 0.94 0.95 0.87 0.88

77 107 131 154 173 59 82 100 117 132 48 66 81 94 106 214 296 363 425 478 164 225

0.56 0.54 0.51 0.52 0.52 0.59 0.49 0.54 0.58 0.61 0.61 0.52 0.56 0.63 0.65 0.55 0.55 0.56 0.58 0.60 0.56 0.56

329 379 393 393 393 334 386 393 393 393 338 390 393 393 393 547 631 655 655 655 556 641

2.38 1.90 1.53 1.32 1.19 3.34 2.32 2.11 1.96 1.83 4.29 3.10 2.71 2.64 2.40 1.41 1.17 1.01 0.90 0.83 1.90 1.60

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R. Hassanli et al. / Engineering Structures 136 (2017) 245–260 Table 8 (continued) Wall name

VFEM (kN)

Failure mode

Proposed approach

MSJC 2013 Flexural

W3.0-3.8-3 W3.0-3.8-4 W3.0-3.8-5 W3.0-4.8-1 W3.0-4.8-2 W3.0-4.8-3 W3.0-4.8-4 W3.0-4.8-5 W4.2-2.8-1 W4.2-2.8-2a W4.2-2.8-3a W4.2-2.8-4a W4.2-2.8-5a W4.2-3.8-1 W4.2-3.8-2 W4.2-3.8-3 W4.2-3.8-4a W4.2-3.8-5a W4.2-4.8-1 W4.2-4.8-2 W4.2-4.8-3 W4.2-4.8-4 W4.2-4.8-5 a

486 520 568 218 320 366 395 430 767 1014 1001 1021 1028 575 757 885 1010 1070 446 609 719 790 855

Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Shear Shear Shear Shear Flexure Flexure Flexure Shear Shear Flexure Flexure Flexure Flexure Flexure

Shear

VEQN (kN)

VEQN/VFEM

VEQN (kN)

VEQN/VFEM

VEQN (kN)

VEQN/VFEM

420 476 526 202 271 320 363 401 662 925 1129 1316 1481 492 679 818 955 1054 394 533 642 734 806

0.86 0.92 0.93 0.93 0.85 0.87 0.92 0.93 0.86 0.91 1.13 1.29 1.44 0.86 0.90 0.92 0.95 0.99 0.88 0.87 0.89 0.93 0.94

276 322 363 134 183 223 261 294 433 590 706 843 942 331 450 539 643 723 272 367 438 524 588

0.57 0.62 0.64 0.62 0.57 0.61 0.66 0.68 0.57 0.58 0.70 0.83 0.92 0.58 0.59 0.61 0.64 0.68 0.61 0.60 0.61 0.66 0.69

655 655 655 563 650 655 655 655 776 890 917 917 917 788 906 917 917 917 800 917 917 917 917

1.35 1.26 1.15 2.58 2.03 1.79 1.66 1.52 1.01 0.88 0.92 0.90 0.89 1.37 1.20 1.04 0.91 0.86 1.79 1.51 1.28 1.16 1.07

Lateral strength using proposed flexural expression is less than the shear expression of MSJC (2013).

1000

the proposed approach, reveals that considering the proposed approach for flexural strength, together with the current shear expression provided in MSJC 2013 (Eq. (19)), yields an accurate prediction of the strength and failure mode.

Proposed Approach- Set I MSJC2013- Set I

VEQN (kN)

800

Proposed Approach- Set II MSJC 2013- Set II

600

5. Conclusions 400 200 0 0

200

400

600

800

1000

VFEM (kN) Fig. 15. Strength prediction of finite element models.

The value of VEQN/VFEM for walls that failed in flexure, calculated ignoring the PT bar elongation (MSJC 2013), varies from 0.43 to 0.85 with an average value of 0.56 and a COV of 19.7% in set I, and varies from 0.49 to 0.69 with an average of 0.59 and a COV of 8.9% in set II. Using the proposed approach, VEQN/VFEM varies from 0.75 to 1.01 and 0.75 to 0.98, with an average of 0.91 and 0.88, and a COV of 19.5% and 5.2%, for set I and set II, respectively. This comparison shows that while ignoring the elongation of PT bars underestimates the flexural strength of PT-MWs, the proposed expression can effectively predict the strength. Furthermore, as elongation of the PT bar is not considered in MSJC, it does not predict the correct mode of failure. For example, wall W4.2-3.8-5 failed in shear while according to the MSJC it is expected to exhibit a flexural mode of failure. Similar findings were observed by Ryu et al. [3]. However, considering the proposed design approach, the predicted flexural strength was found to be higher than the predicted shear strength, calculated using Eq. (19), for the walls that failed in shear, indicating that the recommended design method could successfully predict the behavior and failure mode. In conclusion, comparing the strength and failure mode of the walls presented in Tables 7 and 8, with the results obtained from

This manuscript used finite element models to develop a design expression to predict the in-plane flexural strength of posttensioned concrete masonry walls (PT-MWs). The numerical models of six large scale PT-MWs were developed and validated against experimental results. A parametric study was performed to investigate the effect of different parameters on the behavior and strength of PT-MWs. Multivariate regression analysis was performed to develop an equation to evaluate the rotation of PTMWs at the peak strength. The developed equation was incorporated into the flexural analysis of PT-MWs. The flexural and shear strength of 14 wall specimens collected from the literature and the FEM walls were calculated using the MSJC and the proposed approach. This showed that the proposed approach leads to an accurate and rational evaluation of the flexural strength of unbonded PT-MWs. It is concluded that the proposed expression could significantly improve the strength prediction of PT-MWs and that disregarding the elongation of the PT bars in the unbonded PT-MW results in a highly conservative strength prediction. Moreover, the predicted mode of failure might not be accurate due to low predicted flexural strength. Based on the results of this study, the following design recommendations are proposed: – The axial stress ratio should be kept to not greater than 0.15. For axial stress ratios over 0.15, increasing the applied PT force slightly increases the strength and leads to a brittle mode of failure. – The effective stress in the PT bars after the losses is recommended to be less than 60% of the yield strength of the bars. – An iterative solution using Eqs. (7) and (17) in this manuscript is recommended to estimate the stress developed in the PT bars.

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– The results presented here were obtained for cantilever concrete masonry walls. The axial stress ratios considered in this study ranged from 0.03 to 0.3, and the lengths and heights of the walls ranged from 1.8 m to 5.0 m, and 1.8 m to 4.8 m, respectively. Preliminary investigations indicate that the observations and conclusions derived in this study can effectively be used for design of walls beyond these ranges. They can also potentially be applied to other materials such as clay masonry. However, the accuracy and conclusions must be further examined for walls with parameters beyond the range of the parameters considered in this study. Although in this manuscript equations are proposed for unbonded PT-MWs, investigation by the authors revealed that the recommended equation could accurately predict the strength of post-tensioned concrete walls and can be applied to these members as well [15].

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