Force–displacement behavior of unbonded post-tensioned concrete walls

Engineering Structures 106 (2016) 495–505

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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Force–displacement behavior of unbonded post-tensioned concrete 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 27 March 2015 Revised 18 September 2015 Accepted 22 October 2015 Available online 14 November 2015 Keywords: Post-tensioned Concrete walls Seismic response Force–displacement behavior

a b s t r a c t In this paper the behavior of unbonded post-tensioned concrete walls (PT-CWs) was studied. The accuracy of the existing equations provided in international concrete codes in predicting the in-plane flexural strength of unbonded PT-CWs was investigated using a database of 30 walls. Two recently developed models for unbonded wall members with varying distributions of tendons along the wall length were also considered. An analytical procedure, based on the mechanics of rocking walls and geometric compatibility conditions was employed to characterize the lateral force behavior of unbonded PT-CWs. Using the experimental results, the displacement ductility was determined for the considered walls. According to the results, although the expressions provided by the concrete design codes of the United States (ACI 318-14), New Zealand (NZS 3101) and Australia (AS 3600) to calculate the stresses in tendons of unbonded members were originally developed for beam elements, they resulted in a reasonable strength prediction of wall members. However, the concrete design code of Canada (CSA-A23.3-04) provides a highly unconservative prediction of the flexural strength of unbonded PT-CWs. The presented analytical procedure was able to effectively predict the capacity curve of the tested walls. This research also showed that at low levels of axial stress ratio the ductility of unbonded PT-CWs is highly sensitive to small changes in the level of axial stress. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction During the last three decades comprehensive research has been conducted to understand the response of precast concrete structural components and systems [1–6]. Research has shown that precast unbonded post-tensioned concrete walls (PT-CWs) can display a ductile response while carrying high levels of seismic loads [7]. Unbonded post-tensioning induces self-centering behavior to PTCWs, which reduces residual drifts and limits structural damage to the walls’ toe in the event of an earthquake. This may reduce the repair and downtime costs. Deformations of unbonded PT-CWs during an earthquake event are mainly attributed to the rocking mechanism i.e. rotation of the wall at the base [8]. The rocking mechanism leads to variation in the level of post-tension force in the unbonded tendons due to their elongation, which influences the lateral strength of the wall. Hence, the level of post-tension force should be determined by considering displacement and geometric compatibility conditions ⇑ 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.2015.10.035 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.

rather than the strain compatibility assumptions common in bonded reinforcement [9,10]. The provisions currently provided in concrete standards consider the elongation of PT bars; however, they were originally developed for beam elements [8,11]. Hence, the ability of the codes’ expressions to provide accurate flexural strength prediction in wall members is questionable, mainly due to inherent differences in the behavioral mechanism of beam and wall elements. Compared with beams, walls generally have smaller aspect ratios and are usually subjected to a higher level of axial loads. More importantly, unlike beams, the PT bars in wall members are generally distributed along the wall length [12]. Knowledge of an accurate force–displacement behavior is of high importance in the context of performance based design [13]. However, there are limited studies to date that predict the lateral load behavior of unbonded PT-CWs. Kurama et al. [3] employed finite element analysis and used fiber elements to estimate the force–displacement response of unbonded PT-CWs. Although their method could effectively predict the response, it involved computational difficulties which made it unsuitable for design purposes [14]. Perez et al. [6] proposed a simplified method to predict the nonlinear lateral load behavior of unbonded PT-CWs. In their

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Nomenclature* Aps Aps bw a c cc0 cy d di Ec EPS fc 0 fc fc0 fc j fm 0 fm fps fps i fpu fpy fse fse i fyh hw 00 h Ke Lpl lps

i

pre-stressing steel area pre-stressing steel area of the ith PT bar/tendon thickness of the wall depth of equivalent rectangular stress block neutral axis depth total compression force in concrete at decompression point compression zone length assuming a stress of f py is the PT bar/tendon distance from extreme compression fiber to the centroid of pre-stressing steel distance from the extreme compression fiber to the ith PT bar/tendon concrete elastic modulus Young’s modulus of the pre-stressing steel concrete axial stress concrete compressive strength maximum compressive stress in concrete at decompression point concrete stress at distance xj from the neutral axis masonry axial stress masonry compressive strength PT bar/tendon stress at ultimate flexural strength the ith PT bar/tendon PT bar/tendon stress at ultimate flexural strength ultimate rupture stress of pre-stressing steel yield stress of pre-stressing steel initial stress in the pre-stressing steel after losses effective stress in the ith PT bar/tendon after stress losses confining steel yield strength wall height lateral dimension of the confined core effective yield stiffness plastic hinge length unbonded length of the PT bar, mm

lw N sh T V V0 Vanalysis VEXP VEQN Vy a ,b aKe D D0 Df Di Du Dy

e0

ec ec j ecu emu epu epy ese i

h h⁄

qp qs rps i

u0

wall length, mm gravity load longitudinal spacing of confining steel Total tension force in the wall base Lateral force corresponding to wall rotation h base shear at decompression point Lateral strength of the wall obtained using analytical approach Lateral strength of the wall from experimental study Lateral strength of the wall obtained using design equations effective yield strength equivalent stress block parameters post-yield stiffness wall top displacement due to wall rotation wall top displacement at decompression point total wall top displacement elongation of the ith PT bar/tendon displacement at the peak strength yield displacement maximum compressive strain in concrete at decompression point concrete strain at the extreme compressive fiber concrete strain at distance xj from the neutral axis ultimate concrete compressive strength ultimate masonry compressive strain ultimate strain of pre-stressing steel yield strain of pre-stressing steel initial strain in the ith PT bar after immediate losses wall rotation causes PT bar elongation wall rotation ratio of pre-stressing reinforcement volumetric ratio of the confining steel stress in the ith PT bar/tendon maximum wall curvature at decompression point

* The notations that are not presented in the table are described in the manuscript proposed method the force displacement behavior was estimated using a trilinear idealized curve by calculating the wall’s response at three limit states: effective linear limit, PT bar yield limit, and concrete crushing limit. Although the method was able to predict the response of PT-CWs, this approach has the following limitations:

unbonded PT-CWs and elaborates on an existing procedure to characterize the lateral force behavior of unbonded PT-CWs. The employed analytical approach has been developed based on the mechanics of rocking walls and considers displacement and geometric compatibility conditions. 2. Strength prediction of unbonded PT-CWs

– The lateral load response is idealized by a piecewise linear function including three sections. – Pre-determined neutral axis depths are considered based on the assumption that all PT tendons have yielded. – Strain hardening in PT bars is ignored To overcome the abovementioned limitations, Aealeti and Sritharan [14] developed a simplified approach using a trilinear approximation of the neutral axis depth. However, this method also has the following limitations: the lateral load response is a piecewise linear function including three stages, to define the trilinear variation, the neutral axis is estimated using the neutral axis depth calculated at a drift of 2%, and an equivalent stress block distribution is considered to evaluate the stresses and moments in the concrete. Using experimental results, this paper evaluates the accuracy of design expressions provided by different international standards and researchers to predict the in-plane flexural strength of unbonded PT-CWs. It determines the displacement ductility of

Flexural strength prediction of unbonded post-tensioned walls requires an accurate evaluation of the stresses developed in the PT bars at the wall peak strength. In this section, the accuracy of existing design expressions in predicting the flexural strength of unbonded PT wall members is evaluated based on the available test results. Four international design codes are considered, namely the concrete design codes of the United States (ACI 318-14 [15]), New Zealand (NZS 3101 [16]), Australia (AS3600 [17]) and Canada (CSA-A23.3-04 [18]). Two additional expressions developed recently by Henry [7] and by Hassanli et al. [19] are also considered. 2.1. Concrete design codes of the United States, New Zealand and Australia The ACI 318-14 [15], NZS 3101 [16], and AS 3600 [17] consider the following equations to estimate the tendon stresses, in an unbonded flexural member at the peak strength:

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P

0

f ps ¼ f se þ 70 þ

fc 6 f py ðMPaÞ span-to-depth ratio < 35 100qp

f ps ¼ f se þ 70 þ

fc 6 f py ðMPaÞ span-to-depth ratio P 35 ð2Þ 300qp

ð1Þ

0

where according to ACI 318-14, NZS 3101 and AS 3600, f ps in Eq. (1), cannot exceed f se þ 420, f se þ 420, and f se þ 400, and cannot exceed f se þ 210, f se þ 200, and f se þ 200 in Eq. (2), respectively. For wall members the span-to-depth ratio is considerably less than 35; hence, Eq. (1) must be used.

c¼

The Canadian code CSA-A23.3-04 [18] uses Eq. (3) to determine fps.

f ps ¼ f se þ

8000 X ðd  cy Þ ðMPaÞ lps n

ð3Þ

P

where n ðd  cy Þ is the sum of the distance d  cy for each of the plastic hinges in the span under consideration. For wall members with flexural behavior, the formation of the plastic hinge is limited to the bottom of the wall; hence, Eq. (3) can be rewritten as,

f ps ¼ f se þ

8000 ðd  cy Þ ðMPaÞ lps

ð4Þ

The abovementioned equations assume that the PT bars are lumped at the centroid of the bars, hence, consider a single value for effective depth, d. Therefore, if Eqs. (1)–(4) are employed to evaluate the stress in PT bars, the flexural capacity of the PT-CW can be calculated as,

  a lw a M n ¼ f ps Aps d  þN  2 2 2 f ps Aps þ N a¼ 0 bf c bw 

ð5Þ ð6Þ

3. Existing expressions While assuming a single value for effective depth, d, is a reasonable assumption for beam members, its accuracy for wall elements is questionable. Recently, researchers have attempted to develop design expressions to account for the distribution of the PT bars along the length of unbonded wall members. For concrete members, using finite element models, Henry [7] conducted a parametric study, and provided the following equations to predict fps,

 f ps ¼ f se þ

ecu Eps 8lps

!#  0 0:7 " lps f c f c lw 6 f py ðMPaÞ di  1:36 lw f c abf 0c

ð7Þ

and

fc ¼

f se Aps þ N lw bw

ð8Þ

More recently, Hassanli et al. [19] conducted finite element modelling and a parametric study on unbonded PT masonry walls and developed the following expression to predict the stress of each PT bar at the wall’s peak strength.

f ps i

f ¼ f se i þ 0:11lw þ 3475 m 0 fm

!



emu Eps di lps

c

  1 6 f py ðMPaÞ 0

ð9Þ

where lw is the wall length in mm and f m =f m is the stress ratio. Eq. (9) was developed for walls with lw < 3000 mm.

ð10Þ

abf 0m bw

f ps i is unknown but can be determined by simultaneously solving Eqs. (9) and (10) Substituting the notations for concrete, Eqs. (9) and (10) can be rewritten as,

f ps i ¼ f se i þ 0:11lw þ 3475 P c¼

2.2. Concrete design code of Canada (CSA-A23.3-04)

f ps i Aps i þ N

fc 0 fc

!



ecu Eps di lps

f ps i Aps i þ N

abf 0c bw

c

  1 6 f py ðMPaÞ

ð11Þ ð12Þ

The value of ecu , is considered as 0.003 in this paper. Eqs. (7), (9) and (11) consider variable stresses in the PT bars depending on their locations in the cross section; hence, the following equation is employed to evaluate the flexural strength of PT-CWs:

Mn ¼

X

   a lw a þN f ps i Aps i di   2 2 2

ð13Þ

where

a ¼ ac

ð14Þ

The strength prediction obtained using the expressions of concrete design codes, ACI 318-14 [15], NZS 3101 [16], AS 3600 [17], CSA-A23.3-04 [18] and the expressions proposed by Henry [7] (Eq. (7)) and Hassanli et al. [19] (Eq. (11)) will be evaluated in the remainder of this paper by considering experimental results recently presented by Henry [7].

4. Experimental database Table 1 summarizes a test database of 30 precast concrete specimens that were tested by Henry [7]. The variables of the test walls included the panel dimensions, pre-stressing tendon configuration, and concrete compressive strength. According to Table 1, the walls had heights ranging from 2.2 m to 3.3 m, lengths ranging from 1.0 m to 2.0 m, concrete compressive strengths ranging from 0 25.8 MPa to 41.0 MPa and axial stress ratios, f c =f c , ranging from 0.014 to 0.171. Axial stress, fc, is defined as the total applied axial load, including the initial post-tensioning load, divided by the net cross sectional area. All the walls were erected directly on the strong floor and a bedding material was placed between the wall and strong floor. The PT tendons were anchored at the top of the wall and underneath the strong floor, providing an unbonded length equal to the wall height plus 750 mm. The walls were tested under monotonic lateral displacement and testing was terminated when either localized crushing occurred in the wall toe or when one of the prestressing tendons nearly yielded. A total of eight wall specimens were used, however, as the walls were loaded monotonically, each wall panel was tested in both directions before being inverted and retested, enabling each corner of the panel to be tested once resulting in a total of 32 individual tests [7]. Of all specimens, Tests A1 and A2 were excluded in the database due to their premature failure. In these two walls, no bedding material was used between the wall and the strong floor, and full contact was not achieved at the wall base which resulted in stress concentrations and undesirable responses [7].

5. Evaluation of strength prediction For each wall, the predicted lateral strength was calculated using the following equation,

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Table 1 Wall database [7]. Wall

hw (m)

bw (m)

lw (m)

f’c (MPa)

fc/f’c

fpy (MPa)

No. of tendons

Aps (mm2)

A3 A4 B1 B2 B3 B4 C11 C12 C13 C14 C21 C22 C23 C24 D11 D12 D13 D14 D21 D22 D23 D24 E1 E2 E3 E4 F1 F2 F3 F4

3.0 3.0 3.3 3.3 3.3 3.3 3.0 3.0 3.0 3.0 2.2 2.2 2.2 2.2 3.0 3.0 3.0 3.0 2.2 2.2 2.2 2.2 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0

0.14 0.14 0.14 0.14 0.14 0.14 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2

32.9 32.9 31.7 31.7 31.7 31.7 25.8 27.9 25.8 27.9 28.1 28.1 28.1 28.1 26.0 26.0 26.0 26.0 28.1 28.1 28.1 28.1 41.0 41.0 41.0 41.0 39.5 39.5 39.5 39.5

0.014 0.026 0.019 0.036 0.053 0.078 0.078 0.072 0.171 0.158 0.071 0.071 0.156 0.156 0.059 0.059 0.133 0.133 0.033 0.053 0.088 0.122 0.035 0.049 0.072 0.108 0.039 0.053 0.063 0.087

1050 1050 1050 1050 1050 1050 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580 1580

2 2 1 2 3 3 2 2 3 3 2 2 3 3 3 3 5 5 3 3 5 5 1 2 3 3 3 3 5 5

353.40 353.40 176.70 353.40 530.10 530.10 293.10 293.10 439.65 439.65 293.10 293.10 439.65 439.65 439.65 439.65 732.75 732.75 439.65 439.65 732.75 732.75 146.56 293.10 439.65 439.65 439.65 439.65 732.75 732.75

V EQN ¼

Mn hw

ð15Þ

where Mn was obtained using Eqs. (5) or (13). The predicted lateral strength divided by the maximum lateral strength reported during the experimental work, VEQN/VEXP, is presented in Table 2. As shown in the table, using the provisions of ACI 318-14, NZS 3101 or AS 3600, the value of VEQN/VEXP varies from 0.59 to 0.99 with a range of 0.40, an average of 0.79, a standard deviation of 0.11, and a variance of 0.01. However, CSA-A23.3-04 predicts VEQN/VEXP values varying from 0.89 to 2.18 with a range of 1.28, an average of 1.3, a standard deviation of 0.36, and a variance of 0.13. According to Table 2, using CSA-A23.3-04 results in a very unconservative prediction of the flexural strength of unbonded PT-CWs. 0 According to previous research [19], the axial stress ratio, f c =f c was found to be one of the most important parameters determining the behavior of unbonded PT walls. Fig. 1 presents the relationship between VEQN/VEXP and the axial stress ratio. Bias was measured through the slope of the trend line and the R2 of the data points. A trend line with zero slope, shows an unbiased prediction 0 of strength with respect to parameter f c =f c . As shown in Fig. 1, Eqs. 0 (7) and (11), provide a relatively unbiased prediction of f c =f c . Fig. 1 (b) illustrates that CSA-A23.3-04 provides a very biased prediction 0 of f c =f c , and especially for walls with lower levels of axial stress ratio, it provides a very unconservative flexural strength prediction. Eq. (7) from Henry [7] and Eq. (11) from Hassanli et al. [19] both consider the effect of distribution of PT bars along the length of the wall. Using Eq. (7), VEQN/VEXP varies from 0.68 to 1.1 with a range of 0.42, an average of 0.82, a standard deviation of 0.10, and a variance of 0.01. Using Eq. (11), VEQN/VEXP varies from 0.65 to 1.13 with a range of 0.48, an average of 0.89, a standard deviation of 0.11, and a variance of 0.01. Using Eqs. (7) and (11), no specimen has VEQN/VEXP value P 1.2. Therefore, both Eqs. (7) and (11) provide a

Tendon location, di (mm) d1

d2

d3

d4

d5

187 187 – 94 94 94 94 94 94 94 94 94 94 94 187 187 187 187 187 187 187 187 – 94 94 94 187 187 187 187

– – – – – – – – – – – – – – – – 594 594 – – 594 594 – – – – – – 594 594

– – 500 – 500 500 – – 500 500 – – 500 500 1000 1000 1000 1000 1000 1000 1000 1000 500 – 500 500 1000 1000 1000 1000

– – – – – – – – – – – – – – – – 1406 1406 – – 1406 1406 – – – – – – 1406 1406

1813 1813 – 906 906 906 906 906 906 906 906 906 906 906 1813 1813 1813 1813 1813 1813 1813 1813 – 906 906 906 1813 1813 1813 1813

reasonable prediction of the flexural strength of PT-CWs. However, the percentages of specimens that have VEQN/VEXP values less than 0.80 are 55% and 21% using Eqs. (7) and (11), respectively. Hence, Eq. (7) is too conservative for more than half of the test specimens. 6. Lateral force–displacement response of unbonded PT-CWs 6.1. Stress in unbonded PT bars based on displacement compatibility The stress in an unbonded PT bars in PT-CWs is a function of the wall rotation and the location of the bar within the wall’s cross section. In an unbonded cantilever wall with a flexural mode of failure and a rocking mechanism (Fig. 2), assuming small rotations, the wall’s rotation can be expressed as:

tan h  h ¼

Di di  c

ð16Þ

The different components of Eq. (16) are defined in Fig. 2. Considering the plastic deformation concentrated within the plastic hinge length, the rotation of a concrete wall can be determined as [20],

h ¼

Z

Lpl

ecu c

0

ydy ¼

ecu c

Lpl

ð17Þ

However, the elongation of the PT bars starts to occur only after the wall-footing interface joint opens. Hence, the proportion of the wall’s rotation which provides elongation of the PT bars can be expressed as,

h¼

ecu  e0 c

Lpl

ð18Þ

If the tensile strength of the mortar leveling layer at the wall-footing interface is ignored, the wall’s heel starts to open when the stress in the heel reaches zero i.e. decompression.

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R. Hassanli et al. / Engineering Structures 106 (2016) 495–505 Table 2 Comparison of test results with predicted values. Wall

VEQN /VEXP Hassanli et al. [19]

Henry [7]

ACI 318–14 NZS 3101 AS 3600

CSA A23.3

A3 A4 B1 B2 B3 B4 C11 C12 C13 C14 C21 C22 C23 C24 D11 D12 D13 D14 D21 D22 D23 D24 E1 E2 E3 E4 F1 F2 F3 F4

1.10 0.98 0.65 0.77 0.73 0.79 0.77 0.85 0.93 0.94 0.91 0.91 0.95 0.90 0.88 0.83 1.08 0.90 0.86 1.13 0.83 1.03 0.85 0.76 0.84 0.93 0.97 0.97 0.88 0.92

1.10 0.97 0.72 0.80 0.71 0.72 0.68 0.76 0.77 0.8 0.78 0.78 0.78 0.74 0.79 0.75 0.93 0.77 0.81 0.99 0.71 0.87 0.84 0.74 0.79 0.85 0.94 0.92 0.82 0.85

0.74 0.92 0.71 0.63 0.63 0.72 0.66 0.73 0.86 0.87 0.73 0.73 0.85 0.81 0.74 0.70 0.99 0.83 0.59 0.90 0.69 0.92 0.95 0.68 0.78 0.90 0.85 0.91 0.79 0.87

2.18 2.14 1.15 1.09 1.04 1.05 0.89 1.00 0.95 0.99 1.11 1.11 1.01 0.96 1.43 1.35 1.40 1.16 1.69 2.00 1.27 1.43 1.22 0.98 1.05 1.11 1.72 1.61 1.48 1.41

Min Max Average Std Dev. Var. Range Percentage of specimens having VEQN/VEXP < 0.8

0.65 1.13 0.89 0.11 0.01 0.48 21

0.68 1.10 0.82 0.10 0.01 0.42 55

0.59 0.99 0.79 0.11 0.01 0.40 55

0.89 2.18 1.30 0.36 0.13 1.28 0

Decompression at the wall-footing interface takes place 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. Considering a linear stress–strain relationship in concrete, the absolute maximum concrete compressive strain corresponding to the decompression point, e0 , (Fig. 3), can be obtained using Eq. (19).

e0 ¼

P  2 f se i Aps i þ N lw b w E c

Lpl ¼ 0:11lw þ 3475

fc 0 ðmmÞ fc

ð22Þ

To the best knowledge of the authors, this is the only equation found in the literature to estimate the plastic hinge length in unbonded walls. Assuming that the strain is concentrated within a length equal to the plastic hinge length at any value of wall rotation, h, Eq. (21) can be rewritten as,

f ps i ¼ f se i þ Lpl ðec  e0 Þ

  Eps di 1 lps c

ð23Þ

where ec is the concrete average strain along the plastic hinge length at a drift h. It is worth noting that while the term e0 was ignored when calculating the ultimate strength of PT-CWs i.e. Eq. (11), it was not ignored in Eq. (23). This occurred since the value of e0 is relatively small compared to ecu . For practical construction qffiffiffiffi 0 0 0 parameters (e.g. f c =f c of 10%, f c ¼ 40 MPa, and Ec ¼ 4700 f c ) the value of e0 is equal to 0.0005 which represents 15% of ecu . Hence, e0 can be ignored for simplification when the ultimate strain in the concrete is considered. However, since Eq. (23) evaluates the strain in the tendon for different wall rotations, including small rotations where the concrete axial compressive strains have values close to the value of 0.0005, it is not appropriate to ignore the value of e0 .

6.2. Analytical force–displacement response of PT-CWs

ð19Þ

Substituting from Eq. (18) into Eq. (16), the elongation in the ith PT bar can be determined as follows:

  di Di ¼ Lpl ðecu  e0 Þ 1 c

ð20Þ

Hence, the total stress developed in the ith PT bar can be determined as:

f ps i ¼ f se i þ Lpl ðecu  e0 Þ

6.1.1. Plastic hinge length In reinforced concrete columns having bonded reinforced steel, due to the bond between steel and concrete, the strain, cracks and plastic deformation distribute over a higher length of the column compared to columns having unbonded PT bars [9]. To measure the plastic hinge length experimentally, the strain profile needs to be recorded at different heights of the wall. As reported by Henry [7] in an unbonded wall at the toe region the strain changes rapidly within a relatively small length, which is difficult to measure. Consequently, the plastic hinge length is a function of the type, location and length of the measurement devices and hence, other methods such as analytical methods must be used to evaluate the plastic hinge length [7]. Employing a semiempirical approach, Hassanli et al. [19] proposed an expression to calculate the plastic hinge length for unbonded post tensioned (PT) masonry walls. Using that expression and adopting the notations for concrete, the plastic hinge length can be calculated as follows:

  Eps di 1 lps c

ð21Þ

The neutral axis depth, c, can be calculated using Eq. (12). In order to determine the stress in the tendon using Eq. (21), the value of Lpl , needs to be determined, which is explored in the following section.

For walls having small spacing between tendons and adequate shear reinforcement rocking mechanism controls the behavior of unbonded pots-tensioned walls [21]. Hence, this section focuses on predicting the force–displacement behavior of unbonded PT-CWs controlled by a rocking mechanism i.e. no brittle or local failure such as shear or anchorage failure is considered. The walls considered in this manuscript have symmetrical cross sections and distribution of tendons. In the analytical procedure presented in this study, the stress distribution in concrete is determined as a function of concrete strain rather than an equivalent stress block. Hence, the method can be applied to confined or unconfined concrete. The method was originally developed by Rosenboom and Kowalsky [22] to predict the force–displacement response of unbonded posttensioned masonry walls. In this study, the method is modified and validated for unbonded PT-CWs. The procedure incorporates

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1.40

2.40

1.20

2.00 VEQN/VEXP

VEQN/VEXP

1.00 0.80 R² = 0.16

0.60

1.20 0.80

0.40

0.00

0.05

0.10 fc/f'c

0.15

0.00 0.00

0.20

(a) ACI 318-14, NZS 3101 and AS 3600 1.40

1.40

1.20

1.20

1.00

1.00

0.80 R² = 0.06

0.60

0.80

0.40

0.20

0.20 0.05

0.10 fc/f'c

0.15

0.20

0.10 fc/f'c

0.15

0.20

R² = 0.06

0.60

0.40

0.00 0.00

0.05

(b) CSA-A23.3-04

VEQN/VEXP

VEQN/VEXP

R² = 0.25

0.40

0.20 0.00

1.60

0.00 0.00

0.05

(c) Eq. 7

0.10 fc/f'c

0.15

0.20

(d) Eq. 11

Fig. 1. Comparison of VEQN/VEXP in different design expressions.

Fig. 2. PT-CW before and after deformation.

the plastic hinge length expression proposed by Hassanli et al. [19] (Eq. (22)) and the well-established stress–strain relationships developed by Kent and Park [23] for confined and unconfined concrete. Presented below are descriptions of the analytical procedure.

(I) Force–displacement response at decompression point Taking moments about the wall edge, the base shear corresponding to the decompression limit state can be calculated as follows:

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xj

c

εc j dx

ε

Fig. 4. strain of a typical element in the compression zone. Fig. 3. Stress distribution at decompression at wall-footing interface.

K ¼ 1 þ qs P V0 ¼

f se i Aps i di þ N hw

l  w

2

 cc0

l  w

ð24Þ

3

where cc0 can be calculated as,

cc0 ¼ 0:5lw bw e0 Ec

ð25Þ

Assuming a linear function for the variation of the curvature along the height of the wall and by integrating the curvature function along the wall height, the lateral displacement at the top of the wall at the decompression state is [24]:

D0 ¼

u0 h2w

ð26Þ

3

where,

u0 ¼ e0 =lw

ð27Þ

To determine the lateral behavior of the PT-CW beyond the decompression point, the following iterative procedure needs to be undertaken: 1. Assume a top displacement of D and calculate the corresponding wall rotation, h. (h = D/hw) 2. Assume a value of neutral axis depth, c 3. Calculate the strain in each PT bar, eps i , using Eq. (28).

eps i ¼ hðdi  cÞ=lps þ ese i

ð28Þ

4. Calculate the concrete strain at the extreme compressive fiber, ec , using Eq. (29).

ec ¼ hc=Lpl þ e0

ð29Þ

where Lpl can be determined using Eq. (22). 5. Calculate the stress developed in each PT bar using an appropriate PT bar constitutive model. In this manuscript, an elasto-plastic material model with linear kinematic hardening is considered to model the behavior of PT bars, 8

rps i ¼

> < eps i Eps   epy Eps þ eps i  epy ðf pu  f py Þ=ðepu  epy Þ > : 0

eps i 6 epy epy < eps i 6 epu epu < eps i

ð30Þ 6. Calculate the corresponding concrete stress using Kent–Park stress–strain relationships using Eq. (31) [23]. 8

f c j ðec j Þ ¼

0 > > > < Kf c



2 ec j 0:002K

0 > Kf ½1 > > c

:







ec j

2

0:002K

ec j < 0:002

 Z c ðec j  0:002KÞ 0:002 6 ec j 6 ecp

ð31Þ

ec j > ecp

0 0:2Kf c

where

Zc ¼ h

ecp ¼

3þ0:29f 0c 145f 0c 1000

i

0:5 qffiffiffi00ffi

þ qs 3 4

0:8 þ 0:0015 Zc

h sh

 0:002K

0

ð34Þ

ec j ¼ ðxj =cÞec

ð35Þ

ð32Þ

ð33Þ

fc

where as shown in Fig. 4, ec j is the concrete strain at distance xj from the neutral axis 7. Calculate the total compression force, cc, and the total tension force, T, using Eqs. (36) and (37).

Z

Z c f c j dA ¼ f c j bw dx 0 X 0 X rps i Aps i T¼ Ti ¼ c

cc ¼

ð36Þ ð37Þ

where rps i and f c j can be determined using Eqs. (30) and (31), respectively. 8. If cc ¼ T go to next step otherwise return to step 2. Note that if cc – T, in order to reduce the iterations, the c value for the next iteration round can be taken as equal to c⁄ where

c ¼ c þ

(II) Force–displacement response beyond decompression point

f yh

ðT  cc Þ 0 0:85f c bw

ð38Þ

9. Calculate Df ¼ D0 þ D 10. Take the moment about the neutral axis to calculate the total moment capacity, M, using Eq. (39).

M¼

n X i¼1

rps i Aps i ðdi  cÞ þ Nð0:5lw  cÞ 

Z

0

c

f c j xdA

ð39Þ

where n is the number of PT bars. 11. The terms (Df,V = M=hw ) correspond to a point in the force– displacement curve. To obtain another point, return to step 1. It is worth noting that the presented force–displacement procedure considers in-plane flexural response only. The deformation due to shear, sliding and out-of-plane loading is ignored. In rocking walls, shear and sliding deformations typically have small magnitudes [8] and hence can be neglected. Similarly, in reinforced concrete structures having reinforced concrete diaphragms, outof-plane deformation would be minimal and can be neglected. 6.3. Validation of the proposed analytical model The described analytical procedure was validated using the force–displacement responses of eight unbonded PT-CWs tested by Henry (series E and F in [7]) (Table 1). These test series were considered as they were reported to perform exceptionally well and full even contact was observed between the wall and the plaster bedding layer throughout the tests [7]. It is worth noting that the interface material has significant effects on the response of rocking elements [25]. The calculation of the force–displacement responses was based on the steps provided previously. Note that in the determination of the stress–strain relationship of concrete in order to account for the confinement effect, the value of K in Eq. (31) was calculated to be equal to 1.066 considering the reinforcement mesh used in the wall construction. As reported, four layers of mesh with a diameter of 4 mm, spacing of 150 mm and a yield stress of 475 MPa were used in the construction of the walls [7]. Fig. 5 compares the lateral force–displacement curves determined using the analytical approach and the results obtained during the experimental work. As shown in the figures, the model is

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R. Hassanli et al. / Engineering Structures 106 (2016) 495–505

20

40

60

80

0

100

Displacement (mm)

Analysis EXP 0

20

40

160 120 80 Analysis EXP

40

Lateral Force (kN)

Lateral Force (kN)

200

80

20

30

Analysis EXP

20 0

100

60 40

20

40

80

60

(b)

(c)

Analysis EXP

20 0

0

Displacement (mm)

0

350

350

300

300

160 120 80 Analysis EXP

40

200 150 Analysis EXP

50 0

0

40

250

100

10

20

30

20

30

40

50

250 200 150 100

Analysis EXP

50 0

0

40

10

Displacement (mm)

(d)

200

0 10

40

80

240

0 0

60

Displacement (mm)

(a) 240

60

Lateral Force (kN)

20

100

80

Lateral Force (kN)

0

40

Lateral Force (kN)

Analysis EXP

10

60

Lateral Force (kN)

20

Lateral Force (kN)

Lateral Force (kN)

30

0

100

80

40

10

20

30

40

0

10

20

Displacement (mm)

Displacement (mm)

Displacement (mm)

Displacement (mm)

(e)

(f)

(g)

(h)

30

Fig. 5. Comparison of the experimental and analytical force displacement curves of walls. (a) E1, (b) E2, (c) E3, (d) E4 (e) F1, (f) F2, (g) F3, and (h) F4.

able to correctly predict the wall strength, initial stiffness and rotational capacity. The model can also capture the displacement recorded at the peak strength. The maximum lateral strength of the walls using the force– displacement curves of the analytical approach shown in Fig. 5, VAnalysis, is presented in Table 3. The lateral strength of the walls obtained from the experimental work, VEXP, is also provided in the table for comparison. According to Table 3, the analytical approach could accurately predict the strength of the walls. The predicted strength of the specimens using the analytical approach falls within ±6% of the average of the test results. Compared with the strength predictions using the different methods provided in Table 2, the analytical approach resulted in a better prediction. The reason can be attributed to considering the stress–strain behavior of concrete, rather than assuming an equivalent stress block, and taking into account the kinematic hardening behavior of the PT bars. Figs. 6 and 7 compare the force developed in the PT bars obtained from the experimental work and the analytical approach for series E and F walls, respectively. As shown in Figs. 6 and 7, in general, the analytical model showed a good correlation with the experimental results, but slightly overestimated the force in the bars. Comparing the force developed in each PT bar at the peak strength of the wall using the analytical approach and experimental results, revealed that using the analytical approach, the PT force in the bars at the peak strength was overestimated by 0–9%, with an average of 4%.This overestimation can be attributed to the losses that occurred in the PT bars during testing, which are not considered in the analytical model. These losses occur due to movement of anchorage, deflection of anchorage plates, elastic shortening of the wall, the friction between the PT tendon and the wall of the

Table 3 Prediction of the lateral strength using analytical approach. Walls

VAnalysis (kN) VEXP (kN) VEXP/VAnalysis

E1

E2

E3

E4

F1

F2

F3

F4

36.4 38.4 1.05

68.1 69.9 1.03

88.0 83.1 0.94

96.2 96.9 1.01

195.0 182.4 0.94

207.1 214.7 1.04

306.7 287.2 0.94

325.2 328.4 1.01

ducts at higher drift ratios, and the inaccuracy in the material constitutive models. Fig. 8 compares the neutral axis depth obtained from the analytical approach with that recorded during the experimental work. In general, the analytical model provides a reasonable correlation with the experimental data. However, for small displacements, the analytical approach tends to underestimate the neutral axis depth. As described, the presented analytical method was able to accurately predict the force–displacement behavior of PT-CWs. However, it includes an iteration procedure and computational effort which suits implementation in spread sheets or design software. The process can be simplified for hand calculations with reasonable accuracy. For example, an elasto-perfectly-plastic behavior can be considered for PT bars, and/or a simpler stress distribution can be assumed for concrete. Moreover, the method can be more simplified and less iterative by considering pre-defined values for neutral axis depth [14]. 7. Bilinear idealization of capacity curves The displacement ductility values, l, of the PT-CWs presented in Table 1 were calculated using the following equation [24]:

l¼

Du Dy

ð40Þ

where the Dy and Du values were obtained using the bilinear approximation of the force–displacement response of the walls. The idealized bilinear curves were developed following the procedure provided by FEMA 356 [26]. As shown in Fig. 9, the ultimate displacement, Du , is the displacement at the peak strength, Vy and Dy are the effective yield strength and yield displacement, and Ke and aKe are the effective yield and post-yield stiffness, respectively. An iterative procedure was used to determine the bilinear idealized curves assuming the equal energy concept. An example of bilinear approximation of the load displacement curve of the tested specimens is shown in Fig. 9. As shown in Fig. 5, the unbonded PT walls displayed approximately a bilinear elastic response. Similar bilinear elastic responses were observed for the rest of the wall specimens of the database. In unbonded post-tensioned walls, the nonlinearity

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1400 1200 1000

1100 20 40 60 Displacement (mm)

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0

Analysis

1000 800

1400 1300 1200 1100

T2

1200 1000

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0

1600

T1 T2 T3

1400

20

40

1200 1000 800

1400 1300 1200

20

40

60

0

Displacement (mm)

Displacement (mm)

T1 T2 T3

1500

1100 0

60

10 20 30 Displacement (mm)

Analysis

600 0

1300

20 40 60 Displacement (mm)

1600

800 20 40 60 Displacement (mm)

1400

Analysis

T1

1400

T1 T2 T3

1500

1100 0

60

Tendon Stress (MPa)

Tendon Stress (MPa)

Tendon Stress (MPa)

20 40 Displacement (mm)

1600

T1

0

1200

Analysis

1600 1500

1400

600

800 0

1600

T1 T2 T3

Tendon Stress (MPa)

1400

T2

Tendon Stress (MPa)

1500

1600

T1

Tendon Stress (MPa)

1600

T1

Tendon Stress (MPa)

Tendon Stress (MPa)

1600

10 20 30 Displacement (mm)

Experimental

Experimental

Experimental

Experimental

(a)

(b)

(c)

(d)

Fig. 6. Force in tendons of walls (a) E1, (b) E2, (c) E3, and (d) E4.

Tendon Stress (MPa)

1600

1600

T1 T2 T3

1400

T1 T2 T3

1500 1400

1200

1300

1600 1500 1400

1100

1300

1200

800

1100

600

1000 10 20 30 Displacement (mm)

1100 700 0

10

20

30

1000 0

Displacement (mm)

1600

30

1300

1600 1500 1400

1100

1300

1200

800

1100

600

1000 0

10

20

30

Displacement (mm)

10 20 Displacement (mm)

T1 T2 T3 T4 T5

1700

1300

1000

0

Analysis

T1 T2 T3 T4 T5

1500

1400

1200

20

Analysis

T1 T2 T3

1500

10

Displacement (mm)

Analysis

T1 T2 T3

1400

1200

900

Analysis 1600

T1 T2 T3 T4 T5

1700

1300 1000

0

Tendon Stress (MPa)

T1 T2 T3 T4 T5

1500

1200

900

1100 700 0

10

20

Displacement (mm)

30

1000 0

10

20

30

Displacement (mm)

0

10 20 Displacement (mm)

Experimental

Experimental

Experimental

Experimental

(a)

(b)

(c)

(d)

Fig. 7. Force in tendons of walls (a) F1, (b) F2, (c) F3, and (d) F4.

R. Hassanli et al. / Engineering Structures 106 (2016) 495–505

EXP

800 600 400

EXP

800 600 400

600 400 200

0

0

0 0

30

10 20 Displacement (mm)

1200

Analysis

NA (mm)

800 600 400

800 600 400

0

0 10 20 Displacement (mm)

(e)

1200

30

30

Analysis

1000

EXP

400

0

0

10 20 Displacement (mm)

(d)

600

200

30

0

800

200 10 20 Displacement (mm)

30

Analysis

1000

EXP

200 0

10 20 Displacement (mm)

1200

Analysis

1000

EXP

400

(c)

NA (mm)

1000

600

0 0

(b)

1200

EXP

800

200

NA (mm)

(a)

30

Analysis

1000

EXP

800

200 10 20 Displacement (mm)

1200

Analysis

1000

200 0

NA (mm)

1200

Analysis

1000 NA (mm)

NA (mm)

1200

Analysis

1000

NA (mm)

1200

NA (mm)

504

EXP

800 600 400 200 0

0

10 20 Displacement (mm)

(f)

30

0

10 20 Displacement (mm)

(g)

30

(h)

Fig. 8. Neutral axis depth in, analysis versus experimental, of walls. (a) E1, (b) E2, (c) E3, (d) E4 (e) F1, (f) F2, (g) F3, and (h) F4.

Dri (%)

Lateral force (kN)

100

0

1

Table 4 Characteristics of the bilinearized capacity curves of the test specimens.*

2

Vu 80

αKe

Vy

60

0.6Vy 40

Ke

20

Experiment Idealized response

0 0

Δy

10

20

30

Δu

40

Displacement (mm) Fig. 9. A sample of bilinearization of the capacity curve according to FEMA 356 [26] (wall C14).

occurs when the interface joint at the base of the wall opens significantly, leading to stiffness softening. Hence, the pseudo yielding point does not necessarily correspond to yielding of any bars, but rather to stiffness softening in the wall [27]. Table 4 presents the characteristics of the bilinearized capacity curves and the ductility of the test specimens obtained using Eq. (40). As shown in the table, the considered walls presented a wide range of ductility, ranging between 7.1 and 74.8, with an average of 28.6. Fig. 10 presents the calculated ductility versus axial stress ratio, 0 f c =f c . The general trend of data in Fig. 10 illustrates that the ductility of unbonded PT-CWs is highly dependent on the axial stress ratio. Using the ductility values of the wall specimens provided in Table 4 and considering different wall parameters, multiple regression analyses were performed to find the best fit curve for the ductility values. A regression line with the following expression, having a R2 value of 0.86 and P-value of 0.00, was found to be the best equation for estimating the ductility of the unbonded PT-CWs.

l ¼ 1:7

0

fc fc

ð41Þ

*

Wall

Dy

Du

Vy

Vu

Ke

aK e

l

A3 A4 B1 B2 B3 B4 C11 C12 C13 C14 C21 C22 C23 C24 D11 D12 D13 D14 D21 D22 D23 D24 E1 E2 E3 E4 F1 F2 F3 F4

0.6 2.9 1.5 1.4 1.8 5.1 2.1 2.1 3.0 4.7 1.3 0.8 1.8 2.0 1.3 1.7 2.4 2.7 0.5 1.0 1.2 1.7 1.4 0.9 1.3 2.7 1.1 0.9 1.1 2.1

36.1 35.5 111.5 53.7 54.0 51.9 78.4 64.3 56.3 33.2 30.4 29.8 25.7 25.4 39.5 42.6 24.6 35.1 29.7 20.7 22.9 12.9 86.1 61.4 52.5 35.1 35.3 30.0 28.8 21.8

47.8 86.3 13.4 28.8 39.7 59.1 37.4 36.8 76.0 70.2 45.6 50.5 101.7 102.5 133.3 130.8 213.6 80.6 105.3 121.8 276.2 330.4 28.5 43.3 61.3 80.6 124.1 163.1 194.4 252.7

109.7 124.2 29.4 51.3 70.6 79.0 65.9 60.9 91.2 88.8 81.9 81.8 125.4 125.8 189.7 201.1 297.2 96.9 231.9 214.7 407.6 389.5 38.4 69.9 83.1 96.9 182.6 214.7 287.2 328.4

75 30 9 20 22.5 11.5 18 17.3 25 15 35 62 56 52 99 75 88 30 195 120 235 190 21 49 47 30 113 175 180 120

1.7 1.2 0.1 0.4 0.6 0.4 0.4 0.4 0.3 0.7 1.2 1.1 1.0 1.0 1.5 1.7 3.8 0.5 4.3 4.7 6.1 5.3 0.1 0.4 0.4 0.5 1.7 1.8 3.4 3.8

56.5 12.3 74.8 37.3 30.6 10.1 37.7 30.2 18.5 7.10 23.3 36.6 14.2 12.9 29.3 24.4 10.1 13.1 55.0 20.4 19.5 7.4 63.4 69.5 40.3 13.1 32.1 32.2 26.6 10.4

Units: Dy, Du: mm, Ke, aKe: kN/mm.

Note that as some of the specimens of wall test group A, presented in Table 1, were reported to have immature failure, this group has been removed from the analysis. The correlation between the data set and Eq. (41) in Fig. 10 indicates that in unbonded PT-CWs ductility is inversely proportional to the level of axial stress ratio. This conclusion is similar to the results reported by Caballero-Morrison et al. [28] for slender concrete columns.

Displacement duclity, μ

R. Hassanli et al. / Engineering Structures 106 (2016) 495–505

80

Experiment 1.7(f 'c/fc)

60 40 20 0 0.00

0.05

0.10 fc/f'c

0.15

0.20 0

Fig. 10. Ductility versus axial stress ratio, f c =f c .

8. Conclusions This paper investigates the accuracy of current equations and procedures to predict the in-plane flexural strength of unbonded PT-CWs based on available experimental results. Experimental results were also used to determine the displacement ductility of unbonded PT-CWs. Moreover, this paper elaborated on an existing procedure to characterize the lateral force behavior of unbonded PT-CWs. Within the range of wall configurations and parameters considered in this study, the following conclusions can be drawn: – Although the expressions provided by the concrete design codes of the United States (ACI 318-14 [15]), New Zealand (NZS 3101 [16]) and Australia (AS 3600 [17]) to calculate the stresses in tendons of unbonded members were originally developed for beam elements, they were able to accurately predict the strength of PT-CWs. However, the provisions of the concrete design code of Canada (CSA-A23.3-04 [18]) resulted in a highly unconservative prediction of the flexural strength of unbonded PT-CWs. – This paper showed that two equations developed recently for unbonded PT-CWs (Eq. (7) by Henry [7] and Eq. (11) by Hassanli et al. [19]), in which the distribution of tendons along the wall length is considered, were able to accurately predict the flexural strength. – The analytical approach employed in this paper, developed based on geometric compatibility conditions, accurately predicts the lateral force behavior of unbonded PT-CWs. The wall strength calculated using the analytical approach was within 6% of the experimental results. Moreover, the analytical approach could effectively predict the force developed in posttensioned bars at different drift ratios as well as the wall initial stiffness and rotational capacity. – In unbonded PT-CWs ductility is inversely proportional to the level of axial stress ratio. Eq. (41) is proposed to estimate the ductility displacement of unbonded PT-CWs.

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