Inherent limitations and alternative to conventional equivalent strut models for masonry infill-frames

Engineering Structures 141 (2017) 666–675

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Inherent limitations and alternative to conventional equivalent strut models for masonry infill-frames A. Mohyeddin a,⇑, S. Dorji a, E.F. Gad b, H.M. Goldsworthy c a

School of Engineering, Edith Cowan University, Joondalup, WA 6027, Australia School of Engineering, Swinburne University of Technology, Hawthorn, VIC 3122, Australia c Dept. of Infrastructure Engineering, The University of Melbourne, Parkville, VIC 3010, Australia b

a r t i c l e

i n f o

Article history: Received 8 September 2016 Revised 29 December 2016 Accepted 28 March 2017 Available online 5 April 2017 Keywords: Infill panel Concrete frame Strut modelling Masonry Reinforced concrete Finite element ANSYS OpenSees

a b s t r a c t Past studies have confirmed that the behaviour of an infill-frame can be remarkably different from that of a bare frame. This becomes specifically critical when the structure is under lateral loads such as wind and earthquake. This paper looks into the fundamentals of the most commonly used analytical method for the analysis of such structures, i.e. equivalent strut modelling. It is shown that even though several equivalent strut models have been proposed since the 1950s, none can be considered as a suitable generic tool to represent the behaviour of all infill-frame structures. It is further demonstrated that not only the total width of strut(s), but also the number and location of strut(s) may vary from one infill-frame to the next. It is also shown that even for the same infill-frame the strut properties change at different drift values. A methodology is proposed to develop an appropriate strut model incorporating the material nonlinearities for any given infill-frame. This methodology requires the analytical results of a primary FE model at a micro level to determine the geometric properties of struts. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction In the United States, skyscrapers were designed in the 1900s based on the assumption that there was no structural contribution from the masonry infill panel towards the structural stiffness and/or strength of the building. However, early observations of such structures under wind loading proved the opposite. Cracks which were developed in masonry infill panels demonstrated a significant contribution from infill panels in resisting lateral loads, whereas strain gauges fixed to columns did not register much strain. A substantial difference between the actual stiffness of such buildings and those calculated based on the assumption of no structural contribution from the masonry infill panel came as other evidence. For the Empire State building, for instance, such analyses determined the actual stiffness of the building to be 4.5 times greater than that of the bare frame. Similar observations noting the substantial differences between the behaviour of a bare frame and an infill-frame during past earthquakes have been reported [21,48,50,28]. ⇑ Corresponding author. E-mail addresses: [email protected] (A. Mohyeddin), [email protected]. edu.au (S. Dorji), [email protected] (E.F. Gad), [email protected] (H.M. Goldsworthy). http://dx.doi.org/10.1016/j.engstruct.2017.03.061 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.

The earliest comprehensive published research on infill-frames can be attributed to Polyakov [38] in which the significance of the difference between the structural behaviour of a bare-frame and that of an infill-frame was explained. Experimental research on this subject commenced in the late 1940s and has since been active. Subsequently, this was accompanied by theoretical and computational studies. Even today, the complex structural interaction between the structural frame and masonry infill panel is still being investigated. The level of complexity is such that a parallel research streamline initiated, almost from the start, with the aim of simplifying the actual behaviour of infill-frames. The first simplifying analogy used for the analysis of infill-frames was to take the infill panel as equivalent to one concentric compressive bracing strut between the top of the windward column and the bottom of the leeward column as shown in Fig. 1. Using such an analogy the analysis of a complex composite structure would downgrade to the analysis of a simple braced frame. The possibility of such an analogy, to the knowledge of authors, was first introduced by L I Onishchik as referenced by Polyakov [38] in the late 1930s/early 1940s. Consequently the width of such a strut (denoted by ‘‘d0” in Fig. 1) was first proposed by Holmes [20]. This paper looks into the fundamentals of the most commonly used analytical method for the analysis of infill-frames, i.e.

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Lateral force

d0

Fig. 1. Internal actions in an infill-frame under lateral loading and formation of a compressive diagonal strut.

equivalent strut modelling. It is shown that even though several equivalent strut models have been proposed since the 1950s, none can be considered as a suitable generic tool to represent the behaviour of all infill-frame structures. 2. Background Since 1961, a large number of experimental, theoretical and numerical studies have been carried out to produce an equivalent strut model for infill-frame structures. The geometric properties of such strut(s), viz. the number of struts, total cross sectional area of strut(s), and the location of strut(s), relate to how the infill panel and frame structurally interact, and hence the contact area between the two at different levels of loading/drift. The total cross sectional area of the struts is normally calculated as the product of the calculated width and the nominal thickness of the infill panel. Polyakov [38] gave an estimate of 20–30% of the perimeter of the infill panel to be in contact with the frame (top of the windward column and bottom of the leeward column) after the initial bond between the infill panel and frame is lost. Another early study on the contact length between the infill and frame was conducted by Stafford Smith [44] who provided a range of between 5 and 50% of the frame height (in a square infill-frame) depending on a relative stiffness parameter given in Eq. (1):

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 Em t sin 2h kh h ¼ h 4Ef Ic hI

ð1Þ

where h = the height of the frame; Em = the modulus of elasticity of the infill panel; t = the thickness of the infill panel; h = the angle of the infill panel diagonal to horizontal; Ef = the modulus of elasticity

of the frame members; Ic = the moment of inertia of the column; and hI = the height of the infill panel. Eq. (1), which has been used extensively by other researchers, is a measure of the stiffness ratio of the infill panel to that of the frame when under lateral loading; the higher this value, the longer the contact length. When using strut models, one should note that the models which have been developed based on the experimental results from steel frames have also been applied to infill-frames with reinforced-concrete (RC) frames, and vice versa. The type of masonry material, scale of the specimens, infill-frame aspect ratio (the ratio of the height to length of the infill-frame), the amount of gap between the infill panel and frame, the effects of perforation(s) in the panel, the amount of reinforcement in RC frame members (i.e. ductile and non-ductile frames), the number of storeys, and the number of bays are some of the variables that have been investigated in different studies, e.g. Holmes [20], Stafford Smith [44], Mainstone [28], Crisafulli et al. [13], Crisafulli and Carr [14], Asteris et al. [5] to name a few. Even though, from very early attempts [28] it was observed that the width of equivalent strut(s) may change from one infill-frame to the next, the attempt has always been to develop a generic strut model to be used for the analysis of any infill-frame structure. Some of strut models are used to calculate the initial stiffness (or natural frequency) of the infill-frame only, e.g. Stafford Smith [43], whereas others are used to calculate the ultimate strength e.g. Holmes [20]; but there are only a few that have attempted to replicate the full force-displacement response of the structure e.g. El-Dakhakhni et al. [16]. Regardless of which model is used, another concern that arises when using strut models is that the shear force and bending moment diagrams of the frame members cannot be properly predicted by these models, which has also been discussed by other researchers, e.g. Asteris [4], Crisafulli et al. [13], Asteris et al. [6]. This is because the actual contact length/area between the frame and infill panel cannot not realistically be represented in a strut model, especially when the infill is replaced by a concentric single strut. In an attempt to resolve this issue some researchers proposed multi-strut models, where the masonry panel is approximated by more than one single strut, e.g. Chrysostomou [10], Crisafulli [12], Thiruvengadam [47] and Burton and Deierlein [7]. Many more studies on the equivalent strut modelling of infill-frames can be found in the literature, e.g. Zarnic and Tomazevic [52], Sobaih and Abdin [42], Angel [2], Zarnic [51], Reinborn et al. [39], Saneinejad and Hobbs [40], Crisafulli [12], Madan et al. [27], Al-Chaar [1], Kappos et al. [22], Combescure [9], Karayannis et al. [23], Celarec et al. [8], and Su and Shi [46].

L

177.8 304.8

#2 @ 65

177.8

304.8 177.8

8#4

B

Section A-A A

A

B

152.4 228.6

1422.4

228.6 304.8

177.8

Specimen 8 9 11

#2 @ 75 4#5

Section B-B Fig. 2. Geometric properties of the infill-frames used in Specimens 8, 9 and 11 [29].

L (mm) 2133.6 2133.6 2946.4

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273 [18], ASCE/SEI 41-06 [3] and EN 1998-1:2004 [17]; the basis of such models is the research by Mainstone [28] which is also examined throughout this article. Mohyeddin et al. [33] conducted a series of FE analyses to capture the nonlinear behaviour of RC frames for up to 4% drift. This was followed by nonlinear analyses of infill-frames at a micro level [34–36] whereby the formation, direction and location of principal stresses in the masonry infill panels of two specimens of Mehrabi (1994), namely Specimen 8 and Specimen 9, were scrutinised over a wide range of drift, i.e. 3%. These two infill-frames have exactly the same overall dimensions, with identical frames as shown in Fig. 2; the only major variations between the two infill-frames are a reduced compressive strength of masonry, i.e. 9.5 MPa in Specimen 8, versus 14.2 MPa for Specimen 9, and a reduced thickness of the infill panel for Specimen 8, i.e. 47 mm, versus 92 mm for Specimen 9. Details of the material properties are given in Table 1. Figs. 3 and 4 show the principal compressive stresses in the two infill panels as the lateral drift increases [34]. In both cases, at as early as about 0.2% drift, the load paths are well established (prior to this the gravity loads applied on the beam and columns are dominant). However, despite the similarities between the two structures, one of the main observations is the difference between the numbers of load paths in the two infill-frames; it is clearly possible to identify the formation of three paths in Specimen 8, and two in Specimen 9. Given that strut models are based on an analogy that such load paths in the masonry can be simplified as compressive struts replacing the masonry infill panel, one would consider a three-strut model to best represent Specimen 8, and a two-strut model to represent Specimen 9.

Table 1 Material properties of infill-frames in Specimens 8, 9 and 11 [29]. Diameter (mm)

Type

Yield stress (MPa)

Ultimate strength (MPa)

Reinforcing steel 6.35 12.7 15.9

Plain Deformed Deformed

368 421 414

450 662 662

Modulus of rupture

Tensile split test

Compressive strength

6.8 6.0

3.1 3.5

30.9 29.6

Ultimate strength (MPa)

Strain at ultimate strength

9.5 14.2 11.4

0.0027 0.0026 0.0025

Specimen Concrete (MPa) 8&9 11 Specimen Masonry 8 9 11

A comprehensive study on comparison between different national standards, which include provisions regarding infillframe structures, can be found in [24]. Such provisions may refer to a revised calculation of the natural frequency of the building, lateral load distribution, load reduction factor, influence of openings, and horizontal and vertical irregularities, in order to consider the structural interaction between the frame and masonry panel. In terms of the structural analysis, strut modelling is the most common method applied by most of the standards, such as FEMA

(a)

(b)

w2

(c)

w3

w1 (d)

(e)

(g)

(f)

(h)

Fig. 3. (a) to (h) Principal compressive stresses in the infill panel of Specimen 8 at drift values (%) of 0.0, 0.08, 0.17, 0.25, 0.63, 0.76, 1.01 and 1.73 [34].

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(b)

(a)

(c)

w2 w1

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

Fig. 4. (a) to (n) Principal compressive stresses in the infill panel of Specimen 9 at drift values (%) of 0.0, 0.02, 0.04, 0.06, 0.17, 0.72, 1.06, 1.18, 1.41, 1.65, 1.88, 2.22, 2.32 and 2.53 [34].

The second observation is that as the lateral displacement increases, the number of such load paths, as well as their widths and locations, also change. The change in the width of struts has already been reported in the literature [45], but little attention has previously been paid to the change in their location in the same infill panel. For instance, Fig. 4 shows that the location of the right-hand-side strut remains almost the same at all drift values, even though its width does not. However, the

contact area of the upper side of the left-hand-side strut clearly changes at different drifts. This strut is in contact with the upper side of the column for drift values less than 1.06%, but is pushing against the beam at the drift of 1.41%; again it substantially changes location at 1.88% drift where the contact with the frame is at almost 2/3rd of the column height. These changes occur as the masonry infill softens in regions of high compression.

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Based on the above observations, Mohyeddin et al. [34] suggested that it might be possible to use the results of such FE analyses at the micro level to find the best potential strut model for any individual infill-frame. They suggested that an ‘‘adaptive strut model” would be the appropriate solution to incorporate the change in the configuration and section properties of such struts, not only for different infill-frames, but also for the same infillframe as the drift level increases. Another primary advantage of such an adaptive-strut-model is that it would be better able to capture the internal actions in frame members. However, to develop the properties of such ‘‘adaptive strut models”, the results of a series of detailed FE analyses at a micro level, covering the full range of drift up to the collapse of the structure, must be available first. 3. Significance In order to simplify the idea of ‘‘adaptive strut modelling” explained in Section 2, which requires an algorithm to change the geometric properties of the strut model as the level of drift increases, a simplified but yet ‘‘Case-Specific Strut Modelling” technique is developed in this study. In order to do this, the plots of principal stresses given in Figs. 3 and 4 are used to find the strut properties for the two infill-frames at different drift values. The results from the proposed strut models are then compared with those from experiments as well as from other strut models available in the literature (Table 2). Based on these analyses, it is demonstrated that none of the proposed strut models available in the literature can be generalised to all infill-frame structures due to inherent differences in the behaviour of such structures; such differences can be observed when an analysis at the micro level, similar to those discussed above, is performed. Based on the analyses carried out, it is suggested that a certain drift limit be used to determine the geometric properties, i.e. the number, location and widths of the struts.

Furthermore, a FE model of Specimen 11 of [29] at the micro level is developed and analysed for the first time to find the geometric properties of a suitable strut model for this infill-frame, and to re-examine the method of ‘‘case-specific strut modelling”. The geometric properties of the strut model for this specimen are calculated based on the recommended drift value from the analyses of Specimens 8 and 9. The geometric and material properties of Specimen 11 are given in Table 1 and Fig. 2. 4. Strut modelling As mentioned above, several analyses were conducted using some of the strut models from the literature. In all of the analyses the nonlinear behaviour of the material is considered as explained later in Section 5. Table 2 gives a brief description of the models used for this purpose. Similar to Table 2, Fig. 5 shows the location of struts in some of the multi-strut models from the literature. Crisafulli and Carr [14] suggested three models with one, two and three struts. The separation between struts is defined as a fraction of the contact length between the infill and frame. ElDakhakhni et al. [16] and Chrysostomou [10] suggested threestrut models with the off-diagonal struts positioned at critical locations depending on the plastic moment capacity of the frame members. The strut models proposed by Crisafulli and Carr [14] do not provide the strut width and hence should be used in conjunction with other references when determining the value of the cross-sectional area of strut(s). Figs. 6 and 7 show the force-displacement results from the different strut models given in Table 2 for Specimens 8 and 9. For the analyses where the number and location of struts are based on Crisafulli and Carr [14], the total strut width is calculated based on Holmes [20], and for the analyses where the strut configuration is based on Chrysostomou [10], the total strut width is based on Chrysostomou and Asteris [11].

Table 2 A summary of some of the strut models from literature. Reference

Strut width equationb

Strut cross sectional areas (mm2)a Specimen 8

Specimen 9

Specimen 11

Application/Comments

Holmes [20]

w ¼ 3d d = infill panel diagonal

39,831

78,723

100,329

Ultimate strength. Proposed for square infill-frames but claimed to be valid for h/l between 0.7 and 1.4

Stafford Smith [43]

w ¼ 4d

29,874

59,042

75,247

Stiffness of square infill frames

Stafford Smith [43]

d w ¼ 11

10,863

21,470

27,362

Stiffness of rectangular infill frames when h/l is 5

Mainstone [28]

w ¼ 0:78ðkh hÞ

0:8

d

27,662

47,707

63,376

Mainstone [28]

w ¼ 0:11ðkh hÞ

0:3

d

8378

15,732

20,262

Ultimate strength of the infill-frame with concrete blocks Stiffness of the infill-frame with concrete blocks

Paulay and Priestley [37]

w ¼ 4d

29,874

59,042

75,247

Stiffness

Liauw and Kwan [26]

w ¼ 0:95h cos hðkh hÞ

28,589

51,891

49,379

Stiffness and ultimate strength; independent of h/l

Durrani and Luo [15]

¼ 0:48 sin 2hðkh hÞ d pffiffiffiffiffiffiffiffiffiffiffiffiffi h4 E t 0:1 m d w ¼ 0:32 sin 2h mE Ic hI

23,072

42,597

49,314

Stiffness

17,648

32,582

42,224

Stiffness for brickwork infill panels

19,775

35,939

34,640

Fill-range force-displacement response

35,065

47,486

67,243

Full-range force-displacement response

0:5

0:5

f

0:4

¼ 0:28ðsin 2hÞ0:6 m0:1 ðkh hÞ where:    m ¼ 6 1 þ p6 tan1 IIbc hl     6 1 þ p6 tan1 IIbc tan h

d

Ib = moment of inertia of the beam Chrysostomou and Asteris [11] El-Dakhakhni et al. [16]

w ¼ 0:270ðkh hÞ w¼

ac h ¼

Proposed method a b

0:4

d

ð1ac Þac h cos h

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðMpj þ0:2Mpc Þ 0 tf m

Mpj and Mpc are the plastic moment capacities of the joint and column, respectively Calculated based on FE analysis

Strut cross sectional area is the product of w and t. For definition of some of the symbols refer to Eq. (1).

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Fig. 5. Locations and number of struts proposed by Crisafulli and Carr [14], top, Chrysostomou [5], bottom left, and El-Dakhakhni et al. [16], bottom right.

200

300 200 100

Experimental Strut area: Holmes

0 0

1

2 Drift (%)

3

400

Lateral Load (kN)

Lateral Load (kN)

Lateral Load (kN)

400 300 200 Experimental

100

Strut area: Stafford Smith (d/4)

0 0

4

1

2 Drift (%)

3

Strut area: Mainstone (Strength) 0 1

2 Drift (%)

3

2 Drift (%)

3

4

Strut area: Mainstone (Stiffness) 1

2 Drift (%)

3

4

100 Experimental Strut area: Paulay & Priestley 0 0

1

2 Drift (%)

3

4

300 Lateral Load (kN)

Lateral Load (kN)

Experimental

0

300 200 100

Experimental Strut area: Liauw & Kwan

0 0

1

2 Drift (%)

3

200 100

Experimental Strut area: Durrani & Luo

0

4

0

200

1

2 Drift (%)

3

4

200

100

Experimental Strut locations: El-Dakhakhni Strut area: El-Dakhakhni

0

Lateral Load (kN)

Lateral Load (kN)

100

4

1

200

0 0

Strut area: Stafford Smith (d/11) 0

4

Lateral Load (kN)

Experimental

Lateral Load (kN)

Lateral Load (kN)

100

Experimental

0

200 200

100

Experimental 100 Strut locations: Chrysostomou Strut area: Chrysostomou & Asteris

0 0

1

2 Drift (%)

3

4

0

1

2 Drift (%)

3

4

Fig. 6. Comparison between the experimental results and strut models given in Table 2 for Specimen 8.

As shown in Table 2, the strut cross sectional area for the same structure, when calculated based on different references, can be as much as 5 times different. This indicates that, depending on the properties of a given infill-frame, a specific strut model may or may not be a suitable substitute. Figs. 6 and 7 also support this

observation over the full range of drift, i.e. depending on the strut model used, the force-displacement result can be substantially different. Fig. 6 indicates that the model by Durrani and Luo [15], when incorporating the nonlinear behaviour of material, gives the best match with the experimental results of Specimen 8; how-

A. Mohyeddin et al. / Engineering Structures 141 (2017) 666–675

Experimental Strut area: Holmes 500

Lateral Load (kN)

0 1

2 Drift (%)

3

600

Strut area: Stafford Smith (d/4)

400 200

4

Experimental Strut area: Mainstone (Strength)

400 200

1

2 Drift (%)

3

0

600

1

2 Drift (%)

3

200 100

Experimental Strut area: Mainstone (Stiffness)

200 0 2 Drift (%)

2 Drift (%)

3

4

Experimental 600

Strut area: Paulay & Priestley

400 200

1

2 Drift (%)

3

4

0

1

2 Drift (%)

3

4

400 200

Experimental Strut area: Durrani & Luo

0 1

1

0 0

Lateral Load (kN)

Strut area: Liauw & Kwan

0

0

600

Experimental

400

Strut area: Stafford Smith (d/11)

800

300

4

Experimental 100

4

0

0

200

0 0

Lateral Load (kN)

600 Lateral Load (kN)

Experimental

0 0

Lateral Load (kN)

300

800

Lateral Load (kN)

Lateral Load (kN)

1000

Lateral Load (kN)

672

3

4

0

1

2 Drift (%)

3

4

300 200

Experimental

100

Strut locations: El-Dakhakhni Strut area: El-Dakhakhni

0 0

1

2 Drift (%)

3

4

Lateral Load (kN)

Lateral Load (kN)

400 400 300

Experimental

200

Strut locations: Chrysostomou Strut area: Chrysostomou & Asteris

100 0 0

1

2 Drift (%)

3

4

Fig. 7. Comparison between the experimental results and strut models given in Table 2 for Specimen 9.

ever according to Fig. 7, the ultimate strength predicted by this model for Specimen 9 is considerably larger than that recorded experimentally. In the case of Specimen 9, it is the strut model proposed by Stafford Smith [43] for calculating the initial stiffness of infill-frames with an aspect ratio of 5 that presents the best match with the experimental results, even though the aspect ratio of this specimen is only about 0.67. Therefore, one can conclude that even by incorporating the material nonlinearities in the available strut models, a model which may prove to be suitable for one infill-frame, may not be applicable to another; even if it were, there is no procedure for determining the suitability of a model for a given infill-frame.

5. Case-specific strut modelling Instead of trying to find one generic strut model applicable to all infill-frames, which has been the aim to date, a case-specific strut model is proposed in this study. In this method, for each individual

infill-frame, the number, location and width of each strut can be determined using the results of a FE analysis at a micro level (similar to those presented in Figs. 3 and 4). Two-dimensional equivalent strut models can then be constructed considering the material nonlinearity for both the concrete frame and masonry infill. In this paper, equivalent strut models were constructed using OpenSees. All of the key material properties of the concrete, steel reinforcement and masonry are given in Table 1. ‘‘Concrete01” and ‘‘ConfinedConceret01” material models are used for the concrete cover and concrete core (confined concrete) in the frame members, respectively. These concrete models are based on the (uniaxial) modified Kent-Scott-Park stress-strain relationship [41] and do not consider any tensile strength for the concrete. A ‘‘non-linear beam-column” element is used to model the concrete members (also see Mohyeddin et al. [36] for more details on modelling using OpenSees and Mohyeddin [32] for modelling using ANSYS). The masonry material is defined using the ‘‘Concrete01” material model and the nonlinear stress-strain relationship suggested

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by Hashemi and Mosalam [19]. This material model considers a parabolic equation for the ascending part (Eq. (2)) and a linear relationship for the descending part; this is close to the stress-strain relationship considered by Mohyeddin et al. [34] for their FE analyses:

e

2 0m

e2 þ

0

2f cm

e0m

e

ð2Þ

200

100

Lateral Load (kN)

Lateral Load (kN)

where f0 cm = the ultimate strength of the masonry; e0m = the strain corresponding to f0 cm; e = strain; and r = stress. For each of the infill-frames several analyses were performed. For each analysis, the strut geometric properties (i.e. the number, location and width of struts) were calculated at specific drift values using Figs. 3 and 4. The force-displacement curves associated with these analyses were compared with the experimental results. As part of this series of analyses, some analyses were further carried out using the average of the geometric properties of the struts. This means that the strut geometric properties were taken as the average of the properties at two, three, four or five drift values. Fig. 8 shows the results of only three analyses related to Specimen 8, where the geometric properties of struts were calculated at drifts of 0.25%, 0.63% as well as the average of 0.25% and 0.63%. It is worth noting that struts do not clearly form at any drift value less than 0.25%, i.e. the gravity loads dominate. Also, the experimental results of Specimen 8 show that the ultimate strength of 189 kN occurs at 0.92% drift. This means that the original struts (load paths) formed prior to this drift might have reached their ultimate strength at 0.92% drift and changed direction/location. Such changes can be related to any masonry failure mode, or major damage in the frame leading to relocation of the contact area between the infill and frame. Generally speaking, all of the strut analyses shown in Fig. 8 show a very close match with the experimental results, but the strut model related to 0.63% drift provides the closest. Out of many analyses conducted, Fig. 9 shows the results of three analyses for Specimen 9 where the strut geometric properties were calculated at drift values of 0.17%, 0.72% and the average of 0.17% and 0.72%. It should be noted that struts do not clearly form at any drift value less than 0.17%, and the ultimate strength of the infill-frame, i.e. 291 kN, occurs at 0.5% drift.

Experimental 0.25% drift

0 0

1

2 3 Drift (%)

Fig. 10. Principal compressive stresses in infill-frame Specimen 11 [29] at 0.65% drift.

Similar to Specimen 8, whilst all of the analyses given in Fig. 9 provide a close force-displacement curve to the experimental results, the strut model based on strut properties at 0.72% drift reveals the best match; it captures the initial stiffness and ultimate strength of the infill-frame, and post-failure behaviour up to collapse. In order to further investigate the method presented above, a FE model of Specimen 11 from Mehrabi [29] was created at a micro level using the generic model developed by Mohyeddin [32], and it was analysed up to 0.70% drift. Fig. 10 shows a plot of principal stresses in the infill panel at 0.65% drift. Similar to the other two specimens, the experimental load-displacement curve for this specimen is compared with those from different models constructed using the strut models presented in Table 2 (Fig. 11). It is interesting that the same strut model which gave the best match to the experimental results of Specimen 9, the Stafford Smith [43] model, also gives the best representation for Specimen 11. The aspect ratio for Specimen 11 is 0.5, which similar to Specimen 9, and, as mentioned previously, this is not consistent with the underlying assumption made in this specific strut model (i.e. applicable to infill-frames with aspect ratio of 5). Fig. 12 shows a comparison between the experimental results and the results of the case-specific strut model constructed based on the strut properties at the drift value of 0.65% (Fig. 10). Fig. 12 demonstrates a very good match with the experimental results, confirming the validity of the casespecific strut model in this case.

200

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0

f cm

100 Experimental 0.63% drift

0 0

4

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200

100

Experimental Average of (0.25 & 0.63)% drift

0

4

0

1

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4

Lateral Load (kN)

Lateral Load (kN)

Fig. 8. Comparison between Specimen 8 experimental results and strut models constructed based on FE results [34] at different drift values.

300 200 Experimental

100

0.17% drift

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4

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r¼

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Experimental 0.72% drift

0 0

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3

4

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Experimental

100

Average of (0.17 & 0.72)% drift

0 0

1

2 3 Drift (%)

4

Fig. 9. Comparison between Specimen 9 experimental results and strut models constructed based on FE results [34] at different drift values.

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1200 Lateral Load (kN)

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Holmes

800 400

800

Experimental

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Strut area: Stafford Smith (d/4)

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400 Experimental

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Strut area: Stafford Smith (d/11) 0

0 2 Drift (%)

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Strut area: Mainstone (Strength)

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0 1

3

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2 Drift (%)

3

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2 Drift (%)

3

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Paulay & Priestley

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1

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Experimental Durrani & Luo

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4

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1

2 Drift (%)

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Strut area: Liauw & Kwan

400

4

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100

0

Experimental

3

200

4

Lateral Load (kN)

Lateral Load (kN)

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2 Drift (%)

2 Drift (%)

300

0 0

1

Lateral Load (kN)

0

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300 200 Experimental 100

Strut locations: El-Dakhakhni Strut area: El-Dakhakhni

0 0

1

2 Drift (%)

3

Strut location: Chrysostomou Strut 400 area: Chrysostomou & Asteris 200 0

4

0

1

2 Drift (%)

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4

Fig. 11. Comparison between the experimental results and strut models given in Table 2 for Specimen 11.

Lateral Load (kN)

300 200

Experimental

100

0.65% Drift 0 0

1

2

3

4

Drift (%) Fig. 12. Comparison between Specimen 11 experimental results and the strut model constructed based on FE results at 0.65% drift values.

6. Conclusions Large variations in representative strut properties proposed by different researchers suggest it is difficult to find one generic strut model applicable to all infill-frames. It is shown that even though a given strut model may provide a good match to one set of experimental results, it may present unacceptable results for another. Further, there is no systematic method to determine which model

may be suitable for a given infill-frame. It should be noted that such limitations further apply to the strut models proposed by Mainstone [28], which are the basis of the strut models recommended by some of the main national and international standards and used by practicing engineers (Figs. 6, 7 and 11). In order to solve this issue a case-specific strut model is proposed. In this method the geometric properties of the strut model are determined based on the results of a FE analysis at a micro level. For the first two examples considered in this study, the best results are obtained if the geometric properties of the struts are calculated at a drift level of about 0.7%. The analysis of a third infill-frame also showed that a case-specific strut model constructed using the strut geometric properties determined at this same level of drift gives a very close match to the experimental results. It is, however, too early to generalise this conclusion as in all cases studied here gravity loads were applied on infillframes (which increases the level of masonry confinement by the beam) before the lateral loads were applied. Therefore, this drift value may not be applicable to infill-frames where such a level of confinement is lower or higher.

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For the first two specimens which were analysed here, the strut widths calculated based on Mohyeddin et al. [34] give a ratio of w9/w8 = 47,486/35,065 = 1.35, which almost corresponds to the ratio of (khh)9/(khh)8 = 1.31. However, the strut widths related to specimens 9 and 11 (or 8 and 11 for that matter), have a ratio of w9/w11 = 47,486/67,243 = 0.71, which is distinctly different from the ratio of (khh)9/(khh)11 = 1.09. Therefore, it does not seem that khh is a suitable identifier of the strut properties of an infillframe, especially when the nonlinear behaviour of an infill-frame is considered. Given that structural software with both geometric and material nonlinear modelling capabilities is commonly available, and that the strut modelling concept is still the most effective way of representing an infill-frame structure, the proposed case-specific strut modelling technique is shown to have the advantage of accuracy and simplicity. However, in order to make this method applicable to everyday design procedures, a large number of sensitivity analyses are required to provide a solid classification of infillframes (and hence remove the FE analysis step), and this research will be continued to achieve this goal. References [1] Al-Chaar G. Non-ductile behaviour of reinforced concrete frames with masonry infill panels subjected to in-plane loading [Ph.D.]. University of Illinois at Chicago; 1998. [2] Angel RE. Behavior of reinforced concrete frames with masonry infill walls [Ph. D.]. University of Illinois at Urbana Champaingn; 1994. [3] ASCE, SEI 41–06. Seismic rehabilitation of existing buildings. USA: American Society of Civil Engineers; 2007. [4] Asteris PG. Lateral stiffness of brick masonry infilled plane frames. J Struct Eng 2003;129:1071–9. [5] Asteris PG, Antoniou ST, Sophianopoulos DS, Chrysostomou CZ. Mathematical macromodeling of infilled frames: state of the art. J Struct Eng 2011;137:1508–17. [6] Asteris PG, Giannopoulos IP, Chrysostomou CZ. Modelling of infilled frames with openings. Open Constr Build Technol J 2012;6:81–91. [7] Burton H, Deierlein G. Simulation of seismic collapse in nonductile reinforced concrete frame buildings with masonry infills. J Struct Eng 2014;140: A4014016. [8] Celarec D, Ricci P, Dolšek M. The sensitivity of seismic response parameters to the uncertain modelling variables of masonry-infilled reinforced concrete frames. Eng Struct 2012;35:165–77. [9] Combescure, D. Some contributions of physical and numerical modelling to the assessment of existing masonry infilled RC frames under extreme loading. In: The First European Conference on Earthquake Engineering and Seismology, Sept 3–8 2006 Geneva. [10] Chrysostomou CZ. Effects of degrading infill walls on the nonlinear seismic response of two-dimensional steel frames [Ph.D.]. Cornell University; 1991. [11] Chrysostomou CZ, Asteris PG. On the in-plane properties and capacities of infilled frames. Eng Struct 2012;41:385–402. [12] Crisafulli FJ. Seismic behaviour of reinforced concrete structures with masonry infills [Ph.D.]. University of Canterbury; 1997. [13] Crisafulli FJ, Carr AJ, Park R. Analytical modelling of infilled frame structures–A general review. Bull New Zealand Soc Earthquake Eng 2000;33:30–47. [14] Crisafulli FJ, Carr AJ. Proposed Macro-Model for the analysis of infilled frame structures. Bull New Zealand Soc Earthquake Eng 2007;40:69–77. [15] Durrani, AJ Luo, YH. Seismic retrofit of flat-slab buildings with masonry infills. In: NCEER workshop on seismic response of masonry walls, 1994. NCEER (National Center for Earthquake Engineering Research), 1–3 to 1–8. [16] El-Dakhakhni WW, Elgaaly M, Hamid AA. Three-strut model for concrete masonry-infilled steel frames. J Struct Eng 2003;129:177–85. [17] EN 1998-1:2004. Eurocode 8: design of structures for earthquake resistance. Part I: General rules, seismic action and rules for buildings, Brussels. Berlin: CEN; 2004. [18] EMA 273. NEHRP guidelines for the seismic rehabilitation of buildings, Prepared by Applied Technology Council (ATC). Washington D.C., 1997. [19] Hashemi A, Mosalam KM. Shake-table experiment on reinforced concrete structure containing masonry infill wall. Earthquake Eng Struct Dynam 2006;35:1827–52. [20] Holmes M. Steel frames with brickwork and concrete infilling. Pro Inst Civil Eng 1961;19:473–8. [21] Kafle B, Mohyeddin-Kermani A, Wibowo A. A report on the visit to the region stricken by the Wenchuan Earthquake. Electron J Struct Eng 2008. Appendix B. [22] Kappos AJ, Stylianidis KC, Michailidis CN. Analytical models for brick masonry infilled R/C frames under lateral loading. J Earthquake Eng 1998;2:59–87.

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Further reading [25] Klingner RE, Bertero VV. Earthquake resistance of infilled frames. J Struct Div ASCE 1978;104:973–89. [30] Mohyeddin-Kermani A, Goldsworthy, HM, Gad, EF. The Behaviour of RC Frames with Masonry Infill in Wenchuan Earthquake. In: Australian earthquake engineering society conference, 2008a Ballarat. [31] Mohyeddin-Kermani A, Goldsworthy HM, Gad, EF. A review of seismic behaviour of RC frames with masonry infill. In: Australian Earthquake Engineering Society Conference, 2008b Ballarat. [49] US Army Corps of Engineers. TI 809–04 Seismic Design for Buildings; 1998.