Development of Code-Appropriate Methods for Predicting the

Capacity of Masonry Infilled Frames Subjected to In-Plane Forces

Final Report September 15, 2007

Principle Investigator: R. Craig Henderson Co-Principle Investigator: Richard Bennett

Graduate Student: Charles J. Tucker

Submitted to: National Concrete Masonry Association

Jason Thompson

Executive Summary

Research Introduction and Purpose Buildings that consist of structural steel or concrete frames infilled with unreinforced

masonry are common across the United States and internationally. Infills have two primary advantages: 1) they are easy and economical to construct, and 2) they provide a strong, ductile system for resisting lateral loads. The research described in this report is intended to: 1) translate existing experimental data into analytical methods for predicting the in-plane (IP) stiffness, capacity and structural behavior of various types of infills and 2) formulate and verify (by comparing them with test data and finite element analyses) code approaches which may be used by practicing engineers for the design of new and the analysis of existing infilled frame structures.

Research Approach and Deliverables

The following describes the approach and corresponding deliverables for this research:

1. Data from known experimental research on IP behavior and capacity of infills has been compiled;

2. An accurate linear finite element analysis (FEA) model that replicates the known experimental stiffness results has been developed;

3. An accurate non-linear FEA model that replicates the known experimental strength results has been developed;

4. Published IP analysis techniques that would be suitable candidates for code-prediction of IP stiffness and capacity (e.g., single strut, tri-strut, plastic collapse theory, and others) have been coalesced;

5. The code-acceptable IP analysis techniques have been compared with test data (and the more rigorous finite element results) and ranked as a function of accuracy;

6. The most promising code-acceptable IP techniques have been modified to improve the predictive results; and

7. Code-appropriate equations for both stiffness and strength have been proposed.

Research Results

Deliverable 1: As a first step in developing accurate analysis techniques and code approaches, an extensive database of literature and test results has been developed and compiled with emphasis being given to full-scale testing with static or quasi-static application of loads. In addition, priority has been given to tests conducted on frames infilled with unreinforced masonry (no vertical reinforcement) without openings. Over four-hundred articles, dissertations, and theses related to masonry infilled frames have been collected. These sources have been cataloged into a searchable Microsoft Access database.

Deliverable 2: Linear and non-linear finite element analysis (FEA) models have been developed that accurately replicate the observed experimental behavior of frames infilled with concrete masonry. Data from ten experimental studies (seven with steel frames, three with reinforced concrete frames) has been collected and used to validate the FEA models.

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The linear FEA model displacements have an average of 2.4 percent difference when compared to the experimental displacements. The FEA stress and strain plots clearly show the development of a diagonal strut within the masonry infill. As has been seen in many experimental studies, the diagonal strut carries a majority of the applied force with much of the masonry infill being near zero compressive stress. The von Mises strain plots (Figure ES-1) shows the development of the diagonal strut in the masonry infill as loading progresses.

100 kN 150 kN 200 kN

250 kN 300 kN 342 kN

Figure ES-1. Strut Development in Linear FEA Model

Deliverable 3: The non-linear FEA model has been compared to the experimental

displacements at the first crack load and at the ultimate load. At the first crack load the average percent difference is negative 7.4 percent. At the ultimate load the average percent difference is 4.3 percent. The stress and strain plots from the non-linear FEA model also show the development of a diagonal strut in the masonry infill, similar to those from the linear model; however, an interesting phenomenon can be seen in the von Mises strain plots, the width of the diagonal strut reduces as cracking occurs. Figure ES-2 shows that prior to and immediately following the first major crack developing in the masonry infill the diagonal strut is wide and well defined. After the first crack occurs, the strut width is still well defined; however, its width is slightly reduced. As loading continues and more cracks develop, the strut narrows even more and its effectiveness is reduced. After the ultimate load the diagonal strut is essentially no longer active in resisting the applied lateral loads.

The strut’s reduced effectiveness is also evidenced in the force-displacement plots (Figures ES-3 and ES-4). The stiffness of the masonry infilled frame can be calculated as the slope of the P-delta plot. As Figure ES-4 shows the slope decreases after each crack occurs both experimentally and analytically. Based on the equivalent diagonal strut philosophy the width and effectiveness of the strut must be reduced in order for this to occur. This is especially important when considering to what load level the equivalent diagonal strut philosophy remains valid.

Overall the comparisons of the experimental force-displacement relationships to the FEA force-displacement relationships are appear good. All of the thirty-three experimental

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force-displacement relationships show a significant increase in the displacement or a decrease in the slope (i.e. stiffness of the system) immediately following the first major crack in the infill. Thirty-one of the FEA P-delta plots show the same type behavior following the application of the first crack load.

First Crack Load Second Crack Load

Ultimate Load Ultimate Load + 2kN

Figure ES-2. Strut Degradation in Non-linear FEA Model

Load-displacement comparison for Frame MC2 (McBride 1984)

0

100

200

300

400

500

0 5

Di

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Load-displacement comparison for Frame R2a (Riddington 1984)

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iv

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compared to the experimental data; however, a majority of the methods significantly over-predict the strength of the masonry infilled frame system.

Deliverables 6 & 7: This study culminates in the presentation of two new equations for predicting the stiffness and strength of frames infilled with concrete masonry. Several new equations have been developed and investigated in this study. Three equations were developed simply by modifying existing equations, while six new equations were developed by regressing stiffness on various infill parameters. Equation ES1 is from a regression of λh versus the calculated experimental strut width divided by the diagonal length (wexp/d). Equation ES1 predicts the stiffness of the masonry infilled frame system at the first crack load to within five percent.

−1.15= h)0.25d( w λ (ES1) where: w = width of equivalent strut (mm)

d = diagonal length of infill (mm) λ = characteristic stiffness parameter (mm-1) h = height of frame (on centerlines of beams) (mm)

Figure ES-2 also has the strut width calculated by Equation ES1 overlaid on each of the strain plots from the non-linear FEA model. As seen in the figure, the calculated width of the diagonal strut appears reasonable through the ultimate load. In order to provide the foundation for performance based design, two strength equations have been developed. Equation 39 estimates the strength of the masonry infilled frame system at the first crack load, while Equation 40 estimates the ultimate strength. Equations ES2 and ES3 predict the strength of the masonry infilled frame system to within six percent.

θcos'6.0 wtfP mfc = (ES2) θcos'05.1 wtfP mult = (ES3)

where: Pfc = horizontal component of the diagonal strut capacity at first crack load (kN) Pult = horizontal component of the diagonal strut capacity at ultimate load (kN) fm’ = compressive strength of masonry (MPa) w = equivalent strut width per Equation ES1 (mm) t = thickness of infill (mm) θ = slope of infill diagonal to the horizontal (degrees)

The proposed stiffness and strength equations provide an accurate, yet conservative,

estimation of the behavior and capacity of frames infilled with CMU. In addition, they are theoretically sound and simple to use. As such, use of these equations should allow engineers to take advantage of the easy and economical construction of infills, as well as their strength and ductility in resisting lateral loads.

Conclusions

The linear and non-linear models are a good representation of the behavior of masonry infilled frames up through ultimate load. The FEA displacements and stresses closely match their experimental counterparts.

The proposed stiffness and strength equations appear to provide an accurate, yet conservative, estimation of the behavior and capacity of frames infilled with CMU. In addition, they are theoretically sound and simple to use.

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Table of Contents

EXECUTIVE SUMMARY ......................................................................................................................... ii

Introduction......................................................................................................................... 1Economical Construction 1Structural Effectiveness 1 Lack of Code Guidance 1

Project Objective................................................................................................................. 2 Project Approach and Deliverables .................................................................................... 2 Background Information and Experimental Database........................................................ 3 Description of Experimental Programs............................................................................... 3

Steel Frames Infilled with CMU 4 Reinforced Concrete Frames Infilled with CMU 6

Analytical Modeling ........................................................................................................... 7 Linear model 7 Non-Linear model 9

Existing Analytical Methodologies................................................................................... 11 Stiffness Methods 11 Strength Methods 15

Results............................................................................................................................... 18 Comparison of Linear Model to Experimental Results 18 Comparison of Non-Linear Model to Experimental Results 19 Comparison of Stiffness Equations to Experimental Results 22 Comparison of Strength Equations to Experimental Results 23 Proposed Code Recommendations for Stiffness and Strength 24

Conclusions and Recommendations ................................................................................. 26 Figures............................................................................................................................... 27 Tables................................................................................................................................ 55 References......................................................................................................................... 69 Appendices........................................................................................................................ 77

Appendix A. Literature Review 77 Appendix B. Database Description and Instructions 80 Appendix C. Preliminary Modeling Considerations 82

C.1. Bare Frame Model 83 C.2. Steel Frame Joint Rotational Capacity Study 86 C.3. FE Mesh Discretization Study 89

Appendix D. ANSYS Explanation 93 Appendix E. Description of Existing Stiffness Methodologies 127 Appendix F. Description of Existing Strength Methodologies 132

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Development of Code Methods for Predicting the

In-plane Capacity of Masonry Infilled Frames

Introduction

Buildings that consist of structural steel or concrete frames infilled with unreinforced masonry (see Figure 1) are common across the United States and internationally (Langenbach 1992). Infills have two primary advantages: 1) they are easy and economical to construct, and 2) they provide a strong, ductile system for resisting lateral loads (Henderson et al. 2003; Al-Chaar, et al. 2002; Dawe and Seah 1994; Shing, et al. 1994; Langenbach 1987; Dawe and McBride 1985).

Economical Construction

The construction process for infilled frames is relatively simple. Generally, the bounding frame is erected, and the masonry is laid in the interior. This approach is efficient and provides an economical option for buildings of most any height. Once constructed, the frame serves to support all of the vertical loads (floors and roof), while the infill material resists the lateral loads (earthquake and wind) that are applied to the structure.

Structural Effectiveness

Though infills have been experimentally investigated since the early 1950s (Polyakov 1956; Wood 1958; Stafford-Smith 1962), the test results have, until recently, been sporadic and conducted on a somewhat unrelated array of infill types. Therefore, the conclusions, though positive, were more anecdotal than analytically quantifiable. This lack of data has changed over the last decade as numerous experimental programs, including both small- and large-scale tests, have been conducted (ElDakhakhni 2000; Al-Chaar 1998; Buonopane 1997; Crissafulli 1997; Dunham 1996; Mosalam 1996; Angel 1994; Mehrabi 1994; Seah 1988 and 1998; Thomas 1989). The results of this recent testing show that infills provide significant strength and ductility for resisting various types of lateral loads – even after considerable cracking (Bennett et al. 1996; Henderson et al. 1993; Dawe and Hatzinikolas 1988). Furthermore, recent earthquake performance indicates that existing infilled frames are capable of withstanding significant seismic loads in a ductile fashion (Humar et al. 2001; Flanagan, et al. 1996; Hart, et al. 1994; Langenbach 1992;) even when intended for use as “undesigned” partition walls.

Lack of Code Guidance

Despite their efficiency and performance, the infill material has, historically, been ignored as a part of the lateral force resisting system due to the fact that an industry-accepted design standard does not currently exist. By requiring that the infilled frame be treated as a simple unreinforced wall, current codes encourage the engineer to consider the masonry as dead load or to use ultra-conservative values for its strength (ACI 530-02/ASCE 5-02/TMS 402-02; IBC 2003). As a result infills are not typically considered as a design option, and expensive retrofit of existing infill structures damaged by seismic forces is often mandated but unnecessary (Langenbach 1992).

If infills are indeed an effective and efficient structural system, one must consider why current codes do not provide an appropriate analytical method for their use. There is a two-fold

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reason for this. The first has been the prior lack of experimental data. Fortunately, as mentioned above, this problem has been largely (and recently) corrected by numerous thorough investigations as well as observation under seismic loads. Furthermore, analytical approaches have been proposed. For example, several compression strut methods (Stafford Smith 1962; El-Dakhakhni, et al. 2003) have been suggested as a simplified model for in-plane (IP) behavior. Likewise, arching action techniques have been developed to model out-of-plane (OOP) resistance to inertial or wind loads (Dafnis, et al. 2002; Flanagan and Bennett 1999).

Nevertheless, a second issue remains – the behavior of infilled frames is complex, with two independent materials (i.e., steel or concrete and masonry) resisting applied loads compositely. Many of the proposed analytical methods have been used to predict behavior on a single, or perhaps two or three, test structure(s). However, neither a thorough compilation of the test data nor a broad comparison of analysis methods with the test data has been undertaken. As a result quantifying infill behavior in terms of code guidelines (which may be used by practicing engineers) has yet to be accomplished.

Project Objective The research described in this report is intended to: 1) translate existing experimental

data into analytical methods for predicting the in-plane (IP) stiffness, capacity and structural behavior of various types of infills and 2) formulate and verify (by comparing them with test data and finite element analyses) code approaches which may be used by practicing engineers for the design of new and the analysis of existing infilled frame structures.

Research Approach and Deliverables The following describes the approach and corresponding deliverables for this research:

1. Data from known experimental research on IP behavior and capacity of infills has been compiled;

2. An accurate linear finite element analysis (FEA) model that replicates the known experimental stiffness results has been developed;

3. An accurate non-linear FEA model that replicates the known experimental strength results has been developed;

4. Published IP analysis techniques that would be suitable candidates for code-prediction of IP stiffness and capacity (e.g., single strut, tri-strut, plastic collapse theory, and others) have been coalesced;

5. The code-acceptable IP analysis techniques have been compared with test data (and the more rigorous finite element results) and ranked as a function of accuracy;

6. The most promising code-acceptable IP techniques have been modified to improve the predictive results; and

7. Code-appropriate equations for both stiffness and strength have been proposed.

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Background Information and Experimental Database As a first step in delivering accurate analysis techniques and code approaches, an

extensive database of literature and test results has been developed and compiled with emphasis being given to full-scale testing with static or quasi-static application of loads. In addition priority has been given to tests conducted on frames infilled with unreinforced masonry (no vertical reinforcement) without openings. Over four-hundred articles, dissertations, and theses related to masonry infilled frames have been collected. These sources have been cataloged into a searchable Microsoft Access database. The database includes for each source: all reference citation information, the author’s abstract, and a summary focused on the present research. In addition, several parameters unique to each research project being reported by the article are included in the database: the frame type, the masonry material, the loading direction and methodology, the presentation of data (i.e. figures, tables, etc.), the scale, the research type (experimental or analytical), and the classification of any equations presented. Appendix A provides a more thorough literature review and Appendix B gives instructions for accessing and navigating within the full bibliographic database that accompanies this report.

Description of Experimental Programs As indicated by the above-mentioned database, literally hundreds of articles related to

research on masonry infilled frames have been published, attesting to both the widespread use of infills and the desire among researchers and engineers to quantify their behavior. A majority of the published data focuses primarily, if not entirely, on the experimental results. While a literature review has been done in conjunction with each of these studies, most analytical techniques have been based solely on the data obtained under that particular study. In other words the existing analysis techniques have been validated by only a few data points.

These studies include those conducted on full scale and reduced scale frames. Several different loading schemes were also used including: static, quasistatic, cyclic, and dynamic. Numerous combinations of frame and masonry materials were also investigated. The studies described in the following sections are the ones from which experimental data has been taken for comparison to the existing in-plane methodologies and for validation of the finite element models developed in this study.

Only full size or large scale test programs with static or quasistatic loading schemes are being considered for comparison and validation of the finite element models. Only steel frames with CMU infill and reinforced concrete (RC) frames with CMU infill are being considered for this study.

The experimental data was collected from ten sources. The test specimens consisted of 30 steel-framed infills and three RC-framed infills, which have been divided into several categories (i.e., geometry of the masonry infilled frame, section properties associated with the frame members, material properties, and loading performance) and delineated in Tables 1 through 4. In each of the tables the masonry infill surrounded by steel frames are listed first with the RC frames being the last three rows. Table 1 gives the geometry of the masonry infilled frame which includes the infill height, the story height, the infill length, the infill thickness, and the face shell thickness. Table 2 has the section properties of the frame members including the

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moment of inertia, the area, and the plastic moment capacity for both beams and columns. The material properties for each masonry infilled frame are shown in Table 3. The material properties included are the infill compressive strength, the modulus of elasticity of the infill and the frame, the masonry shear strength, the masonry tensile strength, and the masonry shear bond strength. Table 4 gives the loading performance of each infilled frame including the load and displacement when the first crack occurred, the ultimate load and the associated displacement, and the calculated stiffness at the first crack load. Steel Frames with CMU Infill

John Dawe, University of New Brunswick, had several graduate students (McBride, Amos, Yong, and Richardson) during the mid-1980s who conducted thirty-four full-scale tests on concrete-masonry infilled steel frames subjected to racking loads. The steel frame consisted of 2,800 mm tall W250 x 58 columns rigidly fixed at their bases and a 3,600 mm long W200 x 46 roof beam connected to the columns by semi-rigid moment connections. The columns were oriented for bending about the weak-axis, while the beam was oriented for strong-axis bending. The masonry panels filled the portal space and were constructed using standard 200 x 200 x 400 mm concrete blocks placed with Type S mortar in running bond. The space between the panel and the frame was filled with mortar to ensure intimate contact at the panel-frame interface. Incremental loads were applied to the top of the frame. Figure 2 shows the geometry of the test setup.

All the panels were constructed by locally employed masons and were reportedly of good quality. In laying up the panels the mortar was applied to the face-shells of both bed and head joints, as is typical practice of the construction industry. The first course was laid on two furrows of mortar placed on the base beam. There was no special preparation of this beam other than the removal of any paint on the steel surface. All mortar joints were approximately 10 mm, and the mason used a line strung between columns to keep the panel straight and in alignment with the frame. To finish the joints they were struck flush and tooled to give a concave joint.

McBride (1984) conducted six tests with monotonic loading. McBride’s test Specimen 5 included bent plates to tie the infill panel to the column. Specimen 6 had two bond beams located at the one-third points of the panel. These two tests have been excluded from this study.

Amos (1985) conducted a total of ten tests, seven with monotonic loading and three with load-controlled quasi-static loading. Four of the ten tests (WC3 to WC6) included door size openings and are excluded from this dissertation.

Richardson (1986) conducted twelve tests using load controlled quasistatic loading. Specimens WD1 to WD4 were subjected to 250 pre-crack loading cycles followed by loading to maximum test loads in both directions. Specimen WD7 was loaded in one direction only. Specimen WD5 included an opening, while Specimen WD6 was reinforced along the loaded diagonal. These two tests are excluded from this study. Specimens WD8 to WD12 were confined by a pinned frame. Typical construction practice would prohibit a building from being constructed with fully pinned frames; therefore, Specimens WD8 to WD12 are also excluded.

Yong (1984) performed tests on six specimens under monotonic loading. Specimen 4 was constructed with 20 mm of airspace at the top of the wall. Specimens 5 and 6 were constructed with a tie system similar to the one used by McBride. Specimens 4 to 6 are excluded from this study.

Along with each full scale test was a series of mortar and two-high prism tests to determine the properties of the masonry used as infill. The infilled frames were subjected to an

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in-plane horizontal load at the centroid of the top beam. The horizontal displacement at the point of the applied load, cracking loads, and ultimate load were measured for each test. Additionally, the deflections of the confining frame, rotations of the upper beam-column connections, strains at the column bases, separation along the infill-frame interface, and deformation of the compression diagonal were also measured during each test.

Throughout the tests the load was increased by 22.2 kN increments. At every 22.2 kN an auto-scan of the data acquisition system recorded the deflection at the top of the columns and the strains at the column bases. At every 44.5 kN all the instrumentation was read. The masonry was continuously monitored to determine if cracks had developed. Any cracks detected were highlighted with a marker and labeled with the load causing the crack. The crack pattern was also recorded on the charts. This procedure was continued until a major crack developed, usually with a corresponding drop in load. At this time the load was allowed to stabilize and a complete set of readings were taken. After this the load was increased to the value obtained before the major crack. Readings were taken again. The process continued in 22.2 kN increments until no further increases in load were realized. The test was considered complete when a displacement of approximately 20 mm was obtained at the point of the applied load. The maximum displacement permitted during testing was limited to give an adequate margin of safety against the development of a plastic hinge in the confining frame. Additional details may be obtained from the thesis written by each graduate student.

El-Dakhakhni (2002) investigated the behavior of steel frames infilled with CMU by constructing and testing five specimens. The focus of his research was retrofitting the infill with glass fiber reinforced plastic (GFRP) laminates. Specimen SP-2 was tested without the GFRP laminate under displacement controlled quasistatic loading. The steel frame consisted of 3,000 mm tall W250 x 32.7 columns semi-rigidly connected at their bases and a 3,600 mm long W250 x 32.7 roof beam connected to the columns by semi-rigid moment connections. The columns were oriented for bending about the weak-axis, while the beam was oriented for strong-axis bending. The masonry panels filled the portal space and were constructed using standard 400 x 200 x 150 mm concrete blocks placed with Type S mortar in running bond. Figure 3 shows the experimental setup used for El-Dakhakhni’s specimen SP-3. Frame SP-2 had the same setup as specimen SP-3 with the exclusion of the door opening. The displacement protocol is given in El-Dakhakhni’s dissertation.

Hendry and Liauw (1991) investigated masonry infilled steel frames using brick and CMU infill. The frame consisted of hollow steel tubes 100 x 100 x 8 mm. The 2,575 mm columns were rigidly connected at their bases. The top and bottom beams were 2,800 mm long and were connected to the columns by rigid moment connections. The masonry panels filled the portal space and were constructed using 440 x 215 x 100 mm concrete blocks placed with cement mortar in running bond. A total of six tests were conducted with CMU infill, five of which were reinforced, thus not used in this study. Figure 4 shows the experimental test setup used by Hendry and Liauw. The masonry infilled frame used in this study did not have the vertical reinforcing steel shown in the figure. The load was applied monotonically at the mid-height of the top beam.

Riddington (1984) conducted a series of six tests on CMU infilled steel frames. Two rigid steel frames were used in the study. The weak frame used the British Universal Column section 152 x 152 x 30 UC for the columns and the top and bottom beams. The infill for the weak frame was square and measured 2,710 mm. The strong frame used the British Universal Column section 406 x 140 x 39 UB for each of the frame members. The infill for the strong

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frame was square and measured 2,485 mm. The loaded column was restrained in the vertical and horizontal direction, while the other column was supported on a sliding bearing. The infill walls were formed from 440 x 215 x 100 mm CMU. The load was applied at the mid-height of the top beam in 10 kN increments. Only specimens 2a and 2b are included in this study; the other tests had gaps between the infill and the surrounding frame. Figure 5 shows the experimental setup used by Riddington.

Reinforced Concrete Frames Infilled with CMU

Richard Angel (1994) tested reinforced concrete frames (RC) infilled with various forms of masonry at the University of Illinois at Urbana-Champaign. The purpose of the research was to investigate the loss of out-of-plane strength for unreinforced masonry infilled frames due to in-plane cracking. The loading sequence for each specimen was an initial displacement controlled in-plane load to a predetermined displacement, followed by a uniform out-of-plane pressure.

Figure 6 shows the test configuration for Angel’s RC frames infilled with CMU. The RC frame was constructed in general accordance with the contemporary codes. As can been seen in Figure 6, an additional course of masonry was laid on the top of the connecting beam to simulate the stiffening effect of infill on the upper story. The frame columns were 300 mm square reinforced with eight #7 bars. The connecting beam was a T-beam with the geometry and reinforcing shown in Figure 7.

The CMU infill was constructed from 4 inch and 6 inch wide blocks laid with face shell mortar only. Type N mortar was used with mix proportions 1:1:6 (cement:lime:sand). Both prism tests and mortar tests were conducted to characterize the masonry.

In-plane loads were applied to the infilled frame using two servo-hydraulic actuators. The actuators were mounted on either side of the wall and were attached to the center of the connecting beam. Overall the loading was displacement controlled; however, one actuator was operated under displacement, and the other was operated under force control. This configuration of loading was stated to be for the prevention of inducing torsion to the wall specimen. The loading histogram is provided in Angel’s dissertation.

Francisco Crisafulli (1997) investigated masonry infilled frames at the University of Canterbury, New Zealand. Two frames were infilled with masonry and tested; however, Test Unit 2 had wedge shaped reinforced concrete corners not commonly used in standard construction practice. The ¾ reduced scale frames were intended to represent the lower floor of a two-story structure. The concrete masonry infill was constructed using solid units 230 x 90 x 75 mm. The frame columns were 150 x 150 mm with a connecting beam 150 x 200 mm bending about the strong axis. Figure 8 shows the frame geometry and reinforcing details.

Two hydraulic actuators were attached to the upper beam at quarter points. In-plane quasistatic cyclic loading was applied under displacement control. The loading histogram is provided in Crisafulli’s dissertation.

Haroun and Ghoneam (1997) investigated the use of fiber composites for reinforcing masonry infilled frames. They tested one virgin RC frame infilled with CMU and two other infilled frames that were reinforced with the fiber composite material. The frames were 4 meters in length and 2.8 meters high. The frame beam and columns were 250 x 400 mm with bending about the strong axis. Each frame member was reinforced with eight #6 bars. The infill was constructed using standard 200 x 200 x 400 mm concrete blocks. Figure 9 shows the experimental test setup used by Haroun and Ghoneam.

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The in-plane loads were applied using a displacement controlled hydraulic actuator. The

frame was cycled at each displacement three times. Loading was continued until the frame had lost approximately twenty percent of its maximum strength. Prisms tests and mortar cube tests were conducted to characterize the infill properties.

Analytical Modeling Linear Finite Element Analysis

Background. There have been numerous finite element analysis (FEA) studies conducted on masonry infilled frames; however, these FEA models were limited to small databases of experimental data (Youssef 1994; El-Dakhakhni et al. 2003; Ghosh and Amde 2002). In addition, the existing models did not attempt to mimic the experimental behavior of the infilled frames at the first crack load. This is an extremely important point in the overall behavior of this structural system. While it is true that the majority of masonry infilled frames have significant capacity beyond the first crack load, significant cracking of the masonry infill would be deemed unacceptable to the typical building owner, and the first diagonal crack plays a substantial role in the subsequent behavior of the infill system. For these reasons, investigation of the behavior of the masonry infill at the first crack load has been a priority.

Scope. In addition to behavior at first cracking, the behaviors at maximum load and just beyond maximum load are also of primary importance. This behavior includes the displaced shape of the frame and infill as well as the stress and strain states of the infill. The linear FEA model, described in this section, focuses on the behavior of the masonry infilled frames, both steel and concrete, up to the first crack load. The non-linear FEA model, described in the next section, focuses on behavior through ultimate load.

Structure description. The structures investigated in this study include 30 steel frames infilled with CMU and three reinforced concrete frames infilled with CMU with properties out lined in Tables 1-3. In all but one case the frame is erected and then the masonry wall is constructed tight to the frame members with no gap. The one exception is Crisafulli’s research (1997) where the masonry wall was erected first and then the RC frame was cast against the wall. The interface between the masonry and the frame is typically filled with the same mortar as was used for the CMU wall.

Analysis description/concept. The analysis of masonry infilled frames is complex due to the combination of several distinct materials. The bounding frame can be constructed using steel or reinforced concrete. Though this research focuses on concrete masonry, the infilling may also consist of brick or structural clay tile. The mortar used for the masonry construction can also exhibit a wide cross section of properties depending on numerous factors including, proportions of the constituent materials, type of cement used, and water content. The variation of the materials coupled with the difficulties in the proper treatment of the interface between the frame and the masonry infill create a complex analysis problem.

In order to overcome these difficulties careful consideration was given to the analysis concept used in this FEA model. The apparent linearity seen in the experimental testing indicates that a linear FEA model could give accurate results up to the first crack load.

Finite Element Modeling. The linear FEA model was developed using ANSYS and assumes that the system behaves linearly up to the first major crack load. This assumption is supported by a review of the experimental force-displacement plots which shows that the vast

7

majority of infilled frames tested have a linear displacement plot up to cracking. Other preliminary modeling considerations are discussed in Appendix C. Appendix D provides the complete input files for both the linear and the non-liner models.

The frame members have been modeled using a beam element (ANSYS element BEAM3, Figure 10) that has tension, compression, and bending capabilities. This 2-D element has three degrees of freedom per node, and the material and section properties of each of the frame members were determined from a review of the experimental studies. Appendix C.1 has a full description of the FEA model of the bare frame and its validation. Appendix C.2 describes the steel frame joint rotational capacity study conducted in order to determine the best joint model for the steel frames.

The masonry infill has been modeled using a plane stress element (e.g., ANSYS element PLANE42, Figure 11) that has plasticity, stress stiffening, large deflection, and large strain capabilities. This element has four nodes with two degrees of freedom per node (i.e., translation in the x and y directions). The thickness of the element was set as the face shell thickness from the experimental studies as this seems to be the most likely path for the transmission of lateral forces and stress.

The interface between the masonry infill and the frame members has been modeled using a contact element (e.g., ANSYS CONTAC12, Figure 12) that does not allow the propagation of tensile forces. This element is used between two surfaces that may maintain or break physical contact and may slide relative to each other. This element is capable of compression only in the direction normal to the surface and shear in the tangential direction. It has two translational degrees of freedom at each node. The contact stiffness was set as the modulus of elasticity of the weaker material, in this case, the masonry infill. Frictional forces are calculated using the coefficient of friction between the two materials in contact. The coefficient of friction was set as 0.95 in the linear case in order to take into account the initial adhesion of the masonry to the frame.

The mesh size used for the analysis is approximately 100mm (square) which is equivalent to one-quarter of the length of the standard concrete masonry unit used for the majority of the CMU infill. A mesh sensitivity analysis (i.e., 400mm, 200mm, 100mm, 75mm, 50mm, and 25mm meshes) demonstrated no significant improvement in the model results with a finer mesh. Appendix C.3 provides a description of the mesh sensitivity study.

The load is applied to the upper corner of the bounding frame as was done in the experimental studies. The load is then transferred to the masonry infill by contact between the frame and the masonry. The first and only loading step for this model is the experimentally obtained first crack load.

The boundary conditions for the FEA model match the experimental boundary conditions. For the infilled frames tested at the University of New Brunswick, the frame columns have fixed bases and the supporting beam has fixed supports at one-third points. Figure 13 shows the experimental conditions and their translation into the FEA model.

The frame tested by El-Dakhakhni (2002) at Drexel University also had columns with fixed bases. The masonry infill was supported by a structural floor rather than a supporting beam. Figure 14 shows how the FEA model replicates this condition with a beam supported at each nodal location.

The infilled frames tested by Riddington (1984) had a pin support for the loaded column and a sliding bearing for the opposite column. The FEA model replicates this condition with restraint in the x and y directions for the loaded column and restraint in the y direction at the

8

opposite column base. The supporting beam was attached only at the columns; therefore, no additional boundary condition is necessary in the FEA model. Figure 15 shows the translation of the experimental conditions into the FEA model.

The Hendry and Liauw (1991) frame had support conditions similar to the frames tested at the University of New Brunswick with the exception of the additional supports for the supporting beam, i.e. both columns had fixed bases. Figure 16 shows the experimental conditions and the resulting FEA model.

All of the CMU infilled reinforced concrete frames (Angel, 1994; Haroun and Ghoneam, 1996; Crisafulli, 1997) had fixed bases with the infill being supported on a structural floor. This was replicated in the FEA model with a beam supported at each nodal location as was done for the El-Dakhakhni frame. Figures 17, 18, and 19 show the translation of the experimental conditions into the FEA models for the RC frames.

Brief statement of findings. The linear FEA model used in this investigation replicates the experimental force-displacement curve with reasonable accuracy up to the first crack load for the 30 steel- and three RC-framed infills. The average percent error for the displacement at the first crack load is 2.4 percent. When the absolute value percent error is used the average is 20.9 percent at the first crack load. See Table 5 for a summary of these findings.

Non-Linear Finite Element Analysis

Background. As previously discussed in the description of the linear model used in this investigation, the existing FEA models did not attempt to replicate the rapid displacement associated with major cracks in the masonry infill. This non-linear model expands upon the results of the linear model to more accurately replicate the displacement behavior of masonry infilled frames.

Scope. This non-linear model focuses on the behavior of masonry infilled frames up to and just beyond the experimentally reported ultimate load. Materials included in this analysis are steel frames, reinforced concrete frames, and CMU infill.

Analysis description/concept. The same complexities discussed with respect to the linear model apply to the non-linear model as well. The non-linear FEA model attempts to overcome these difficulties by taking into consideration the material non-linearities and the structure’s behavior immediately following the development of a major crack in the infill. Conceptually, as the loading increases on the masonry infilled frame, the deformation of the frame and infill increase. For the given frame materials and expected displacements, the stress and strain of the frame members is minimal. However, the stress and strain levels in the masonry infill often result in cracking or crushing of the masonry. The FEA model utilizes the strain levels in the masonry as a predictor of damage and reduces the stiffness of elements that are at high strain levels. The reduction in stiffness of individual elements is achieved by eliminating these highly strained elements and results in an accurate representation of the rapid increase in displacement that occurs after a crack is formed in the masonry. Each stiffness reduction is applied to the highly strained elements at the end of each major load step.

Since the linear FEA model has been shown to be an accurate behavior indicator of localized masonry failure (and no strain information was reported by the experimental investigators), the strain level used as the “kill strain” is the maximum strain from the linear model. This effectively ties the finite element analysis to the experimental studies in a number of ways. First, the experimental infill strains were not reported for the majority of the frames

9

tested. Second, the linear FEA model accurately represents the behavior of the masonry infilled frames up to the first crack load. This coupled with the lack of experimental strains makes the linear model the best indicator of the strain level at which major cracking occurs.

Model assumptions. The same assumptions used in the development of the linear model have also been used in the development of this non-linear model. An additional assumption is that the actual stress-strain curve for the masonry infill can be approximated by a modification of a tri-linear stress-strain curve suggested by El-Dakhakhni et al. (2003). The tri-linear stress-strain curve shown in Figure 20 is a good approximation of the parabolic stress-strain distribution historically used. Again, the performance of the FEA model as compared to the experimental data validates this assumption.

Elements. The discretization, element types, and mesh size are the same as those used for the linear model discussed in the previous section. The modulus of elasticity for the PLANE42 element used in this model is now determined from the slope of the tri-linear stress-strain curve rather than the constant modulus used for the linear model.

When forces are applied to a masonry infilled frame the infill is loaded in a direction parallel to the diagonal. The material properties of the masonry vary with the angle of loading. This inclination results in the modulus of elasticity being reduced by approximately twenty percent for the aspect ratios of the experimental frames used in this study (El-Dakhakhni et al. 2003). Hamid et al.(1987) suggested that the modulus of elasticity can be approximated as 650 times the masonry’s compressive strength in the same loading direction. With the twenty percent reduction the inclined modulus of elasticity is then 520 times the compressive strength of the masonry perpendicular to the bed joints.

Seah (1998) suggested that the compressive strength parallel to the bed joints is approximately seventy percent of the compressive strength perpendicular to the bed joint. Using a geometric averaging technique (in which the square root of the product of the compressive strengths in each orthogonal direction is used as the inclined compressive strength) results in the compressive strength along the incline of 0.84f’m.

The first segment of the tri-linear stress-strain curve shown in Figure 20 is then determined by calculating the strain that corresponds to the modulus of elasticity and a compressive stress of 0.84f’m. The horizontal segment is defined by a stress level of 0.84f’m. The first strain for the horizontal segment is the ending strain calculated for the first inclined section. The ending strain for the horizontal section of the tri-linear stress-strain curve is defined by adding the peak strain to the strain calculated for the first segment. El-Dakhakhni et al. (2003) suggested that the modulus of elasticity of the masonry at peak load is equal to one-half of the initial modulus of elasticity. The final descending portion of the tri-linear stress-strain curve is defined by setting the ultimate strain at 0.01.

Loading. The load is applied to the upper corner of the bounding frame as was done in the experimental studies. The load is then transferred to the masonry infill by contact between the frame and the masonry. The total load has been applied to the infilled frame in several steps. The first loading step is to apply the experimentally obtained first crack load. The next experimentally obtained load is the ultimate load. The difference between the ultimate load and the first crack load is applied in two load steps with half of the difference being applied in each step. The experimental force-displacement plots show that roughly one-third of the frames tested had at least one additional major crack with an associated rapid displacement increase after the first crack load and prior to the ultimate load. This experimental behavior provided the rationale for splitting the load into two load steps between the first crack load and the ultimate load.

10

The final load applied to the FEA model is 2kN of additional load above the ultimate

load. This additional load is simply to allow the structure to stabilize after considering damage at the ultimate load level. As was discussed above, elements that exhibited high strain levels were selected and eliminated for stiffness reduction after each major load step. The boundary conditions for this FEA model are the same as those used in the linear model.

Brief statement of findings. The non-linear FEA model used in this investigation replicates the experimental force-displacement curve with reasonable accuracy. The average percent error for the displacement at the first crack load is -7.4 percent. When the absolute value percent error is used the average is 23.0 percent at the first crack load. This percent error differs from the linear model due to the use of the tri-linear stress strain curve in which the first segment defines a lower modulus of elasticity. The average percent error for the displacement at ultimate load is 4.3 percent with an absolute value percent error average of 37.9 percent. See Table 6 for a summary of these findings.

Existing Analytical Methodologies Numerous techniques exist for the in-plane analysis of the strength and stiffness of

masonry infilled frames. This chapter provides a discussion of the most pertinent of these techniques. The current analytical models normally fall within two classifications, macro-models and micro-models. Macro-models simulate the infill’s contribution (particularly stiffness) to the overall response of the structure. Micro-models focus on the infill behavior specifically (i.e., stiffness, capacity, and load-deflection data), typically as a finite element analysis. The individual macro-analysis methodologies will be presented with emphasis on overall building design (i.e., numerically defining the infill’s role in the response of the structural system to lateral loads).

Stiffness Methods

The stiffness of masonry infilled frames has been the focus of numerous analytical and experimental studies over the last five decades. Many of these studies concluded that the most accurate representation of the stiffness of the masonry infilled structure is that of a braced frame with the masonry replaced by an equivalent diagonal strut. Polyakov (1956) was the first to recognize this behavior and quantify the width of the equivalent diagonal strut. Many of the stiffness methods presented in this section are focused on determining the width of the equivalent diagonal strut. Once the width of the strut is calculated, a simple braced frame analysis is conducted to determine the stiffness of the system. Specifically, the stiffness is based on the elastic shortening of the diagonal strut. The following analytical methods have been investigated and compared to the experimental data, along with several other methods described in Appendix E. The methods shown here (as opposed to those listed in Appendix E), have been chosen for discussion and explanation because they are the five most accurate predictors of the known experimental behavior. The exception to this is Stafford-Smith and Carter’s approach (1969), which provides less accurate results, but establishes the foundational strut width principles used by subsequent researchers.

11

Stafford Smith and Carter, 1969. Stafford Smith and Carter continued the

development of the equivalent strut theory following Polyakov (1956) and Holmes (1961). In their formulation, the effective width of the strut is a function of the relative stiffness of the column and the infill, the length-height proportion of the infill, the stress-strain relationship of the infill material, and the value of the diagonal load acting on the infill. The length of contact, α, between the infill and the frame is given by:

hh λπα

2= (1)

where: α = length of contact (mm) h = height of column between centerlines of beams (mm) λ = characteristic stiffness parameter (mm-1)

In the previous equation, λh is a non-dimensional expression for the relative stiffness of the frame to infill, where λ is defined as:

4'42sin

EIhtEI θ

λ = . (2)

where: λ = characteristic stiffness parameter (mm-1) EI = Young’s modulus of infill (kN/mm2) t = thickness of the infill (mm) θ = slope of infill diagonal to the horizontal E = Young’s modulus of the column (kN/mm2) I = moment of inertia of the column (mm4) h’ = height of the infill (mm)

Stafford Smith and Carter presented a series of curves showing the relationship between the width of the diagonal strut and the stiffness parameter, λh. Jamal et al. (1992) attribute Equation 3 which gives the strut width as “a” to Stafford Smith and Carter.

λπ2

=a (3)

where: a = width of equivalent strut (mm) λ = characteristic stiffness parameter (mm-1)

While this particular method did not rank as one of the better methods, it does provide an equation for the relative stiffness of the column and the infill which is used in numerous other methodologies.

Mainstone, 1971. Mainstone’s work was based mainly on the experimental testing of

small scale micro-concrete infill and scaled brick masonry. He also used the relative stiffness parameter, λ, defined by Equation 2. Similar to Stafford Smith and Carter (1969), a series of best-fit curves was utilized to show the interaction of the diagonal compression strut width and the relative stiffness of the infill and frame. Two sets of equations were presented based on the value of the product λh; one set for brick infill, the other for concrete infill. Equations 4 and 5 are for λh values between four and five. Equations 6 and 7 are for λh values greater than five.

( ) brickhw

wh

eK 4.0'

175.0'

−= λ (4)

( ) concretehw

wh

eK 4.0'

115.0'

−= λ (5)

12

( ) brickhw

wh

eK 3.0'

16.0'

−= λ (6)

( ) concretehw

wh

eK 3.0'

11.0'

−= λ (7)

where: w’eK = effective width of infill considered as a single diagonal strut

for stiffness (mm) w’ = diagonal length of infill (mm) λh = relative stiffness parameter (mm-1) h = height of frame (on centerlines of beams) (mm)

Decanni and Fantin, 1986. Dacanni and Fantin (1986) developed two sets of equations

based on the infill condition; i.e., uncracked or cracked, and the value of the dimensionless relative stiffness factor, λh. Crisafulli (1997) reported these equations as follows for an uncracked panel and a cracked panel, respectively:

85.7085.0748.0≤⎟⎟

⎠

⎞⎜⎜⎝

⎛+= hm

h

ifdw λλ

(8)

85.7130.0393.0>⎟⎟

⎠

⎞⎜⎜⎝

⎛+= hm

h

ifdw λλ

(9)

85.7010.0707.0≤⎟⎟

⎠

⎞⎜⎜⎝

⎛+= hm

h

ifdw λλ

(10)

85.7040.0470.0>⎟⎟

⎠

⎞⎜⎜⎝

⎛+= hm

h

ifdw λλ

(11)

where: w = width of equivalent strut (mm) λh = dimensionless stiffness parameter = λh per Equation 2 dm = diagonal length of infill (mm)

Bennett, Flanagan, Adham, Fischer, and Tenbus, 1996. Bennett et al. (1996)

proposed that a piece-wise linear strut width could be calculated by:

θλπcosC

w = (12)

where: w = width of equivalent strut (mm) C = empirical constant based on infill damage λ = characteristic stiffness parameter (mm-1) θ = slope of infill diagonal to the horizontal

The empirical constant, C, is based on the in-plane displacement and typical infill damage. The values of C vary with frame type and infill material. Bennett et al. (1996) published values of the parameter C for steel frames with structural clay tile infill, steel frames with concrete masonry infill, concrete frames with concrete masonry, and concrete frames with brick masonry.

El-Dakhakhni, Eagaaly, and Hamid, 2003. El-Dakhakhni, Eagaaly, and Hamid (2003)

were unsatisfied with a single equivalent strut for replacing the infill. They believed a three-strut

13

model better represented the actual distribution of forces from the infill to the frame. The total strut area is calculated by multiplying the width given in Equation 13 by the infill thickness.

( )θαα

cos1 hw cc−

= (13)

( )

htf

MMh

m

pcpjc 4.0

'2.02

0

≤+

=−

α (14)

where: w = width of equivalent strut (mm) αc = ratio between the column contact length and its height h = column height (mm) θ = slope of infill diagonal to the horizontal Mpj = minimum of the column’s, the beam’s or the connection’s

plastic moment capacity, referred to as the plastic moment capacity of the joint (kN-mm)

Mpc = column plastic moment capacity (kN-mm) t = thickness of the infill (mm) f’m-0 = masonry strength parallel to bed joints (kN/mm2)

The strut area is then split among the three struts. An upper strut connects the upper beam with the leeward column with one-fourth of the total area, a middle strut connects the two loaded corners with one-half of the area, and a lower strut connects the windward column with the lower beam with the remaining quarter of the area. The two off-diagonal struts are stated to provide a more realistic distribution of the forces between the infill and the frame members.

Equation 14 is the contact length of the loaded infill with the column. Additionally, the beam contact length needed to analyze the infilled frame is shown by Figure 21 and given by Equation 15.

( )

ltf

MMl

m

pbpjb 4.0

'2.02

90

≤+

=−

α (15)

where: αb = ratio between the beam contact length and its span l = length of infill (mm) Mpj = minimum of the column’s, the beam’s or the connection’s

plastic moment capacity, referred to as the plastic moment capacity of the joint (kN-mm)

Mpb = beam plastic moment capacity (kN-mm) t = thickness of the infill (mm) f’m-90 = masonry strength perpendicular to bed joints (kN/mm2)

Proposed MSJC, 2007. The Masonry Standards Joint Committee (MSJC) is currently

developing code provisions for the analysis and design of masonry infilled frames. (The author is the Secretary of the Infills Subcommittee of the MSJC.) The current unpublished working draft is proposing Equation 16 for calculating the width of the equivalent diagonal strut which is a modification of Bennett et al.’s (1996) equation.

θλ cos3.0

=w (16)

where: w = width of equivalent strut (mm) λ = characteristic stiffness parameter (mm-1)

14

θ = slope of infill diagonal to the horizontal

Equation 16 simplifies the “π/C” portion of Equation 12 to a simple factor of 0.3. This factor corresponds to a value of approximately 10.5 for the empirical constant “C.” According to the tables published by Bennett et al. (1996), one would expect significant damage to the masonry infill for this value of “C.”

Strength Methods

The strength of a masonry infilled frame has also been investigated heavily over the last

several decades. Numerous failure modes as well as design philosophies have been presented for estimating the capacity of infilled frames. Several include an equivalent diagonal strut, while others are based on shear friction within the masonry or a plastic analysis that includes frame contributions. Some methods are for the capacity up to the first crack load, while others are for the ultimate load capacity; a few include both capacities. As with the stiffness methodologies, several more strength methods were investigated that are listed in Appendix F.

Klingner and Bertero, 1976. In developing an equivalent strut element for use in the

general analysis program ANSR-I, Klingner and Bertero (1976) defined the mechanical loading characteristics of the strut in several regions of behavior. The first region is the elastic loading region where the axial force in the strut is defined by Equation 17.

vL

EAS = (17)

where: S = axial force in the strut (kN) E = Young’s modulus of the infill material (kN/mm2) A = the cross-sectional area of the strut (mm2) L = length of the strut (mm) v = axial deformation in the strut (mm)

The cross-sectional area of the strut is calculated as the product of the panel thickness and the width of the strut as calculated by Equation 5 discussed earlier.

The second region defined was the strength envelope curve with a strength degradation parameter. This section is represented by Equation 18.

(18) )( vc eAfS γ=

where: S = axial force in the strut (kN) A = the cross-sectional area of the strut (mm2) fc = the compressive strength of the infill material (kN/mm2) γ = strength degradation parameter = 1 for all analysis v = axial deformation in the strut (cm)

Following this region is an elastic unloading section which is then followed by a tension curve defined by Equation 19.

(19) tAfS =where: S = axial force in the strut (kN)

A = the cross-sectional area of the strut (mm2) ft = tensile resistance of the panel reinforcement (kN/mm2)

From a practicality standpoint, this tensile capacity was taken as zero.

15

Flanagan, 1994. Flanagan (1994) presented a modification of Holmes’ (1961) equation

(see Appendix F) for the ultimate capacity of an infilled frame. Both equations consider the strength of the frame and the infill as simply additive.

θδ cos'63 mu Af

hEIP += (20)

where: Pu = ultimate capacity of an infilled frame (kN) E = Young’s modulus of the column (kN/mm2) I = moment of inertia of the column (mm4) h = height of infill (mm) δ = column in-plane displacement taken at beam centerline (mm) A = the cross-sectional area of the strut (mm2) fm’ = compressive strength of masonry (kN/mm2) θ = slope of infill diagonal to the horizontal

The cross-sectional area of the strut is calculated as the product of the panel thickness and the width of the strut as calculated by Equation 12 discussed earlier.

Mehrabi, Shing, Schuller, and Noland, 1996. Mehrabi et al. (1996) evaluated the

lateral resistance of an infilled frame based on a sliding shear failure. Any vertical load is distributed between the frame and the infill based on their axial stiffness. Equation 21 is based on Mohr-Coulomb failure criteria.

www PAF 9.0345.0 += (21) where: Fw = sliding shear strength (kN)

Aw = horizontal cross-sectional area of the infill (mm2) Pw = vertical load acting on the infill (kN)

The constants in the above equation are the cohesive stress from testing and a coefficient of friction, respectively.

Flanagan and Bennett, 1999. Flanagan and Bennett (1999) presented two empirical

equations for the strength of the masonry infill. Equation 22 gives the cracking strength as a function of the masonry tensile strength. The tensile strength is estimated as the square root of the compressive strength.

1000'mcr

crfLtK

H = (22)

where: Hcr = cracking strength (kN) Kcr = empirical constant L = infill length (mm) t = thickness of the infill (mm) fm’ = compressive strength of masonry (MPa)

Based on testing of structural clay tile, a mean value of 0.066 is recommended for Kcr. The second equation given by Flanagan and Bennett is for estimating the corner crushing

load for the infill. The corner crushing strength does not seem to be affected by the frame properties; therefore Equation 23 is based solely on the properties of the masonry.

1000'mult

ulttfKH = (23)

where: Hult = corner crushing strength (kN)

16

Kult = empirical constant t = thickness of the infill (mm) fm’ = compressive strength of masonry (MPa)

FEMA 306, 1999. FEMA 306, “Evaluation of Earthquake Damaged Concrete and

Masonry Wall Buildings: Basic Procedures Manual,” presented four failure modes for masonry infilled frames: sliding-shear failure, compression failure, diagonal tension failure of the panel, and general shear failure of the panel.

The sliding-shear failure is based on the Mohr-Coulomb failure criteria which reduce to Equation 24 for infill panels.

NtLV yi

slide µφσ == infinf)tan( (24) where: Vi

slide = initial sliding shear capacity (kN) σy = vertical stress (kN/mm2) φ = angle of sliding friction of the masonry along a bed joint Linf = length of infill (mm) tinf = thickness of infill (mm) µ = coefficient of sliding friction along the bed joint N = vertical load in panel (kN)

If deformations are small, the sliding shear capacity from Equation 24 becomes zero since the vertical stress is typically only from the panel self-weight. However, if the deformations are significant, the bounding frame, particularly the columns, induces a vertical load due to shortening of the panel. Taking the vertical shortening strain into account, Equation 24 becomes

2infinf θµ m

islide EtLV = (25)

where: Vislide = initial sliding shear capacity (kN)

µ = coefficient of sliding friction along the bed joint Linf = length of infill (mm) tinf = thickness of infill (mm) Em = modulus of elasticity of the masonry infill material (kN/mm2) θ = interstory drift angle = horizontal displacement divided by

story height The equation for the compression failure mode is a modification of one suggested by

Stafford-Smith and Carter (1969). The horizontal shear force in the diagonal strut is calculated from Equation 26.

θcos' 90inf mc fatV = (26) where: Vc = horizontal component of the diagonal strut capacity (kN)

a = equivalent strut width per Equation 5 (mm) tinf = thickness of infill (mm) f’m90 = expected strength of masonry in the horizontal direction,

which may be set at 50% of expected stacked prism strength (kN/mm2)

θ = slope of infill diagonal to the horizontal Diagonal tension failure of the panel is governed by Equation 27 with the masonry

cracking capacity, σcr, as a function of the orientation of the principal stresses in relation to the bed joints.

17

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

inf

inf

inf

inf

inf224.25

Lh

hL

tV crcr

σ (27)

where: Vcr = cracking shear capacity (kN) 25.4 = conversion factor for constant 2√2 (mm/in) tinf = thickness of infill (mm) σcr = cracking capacity of masonry, which may be set at f’m90/20 in

the absence of test results (kN/mm2) Linf = length of infill (mm) hinf = height of infill (mm)

The initial and final general shear failure of the panel is governed by Equations 28 and 29, respectively. The constants at the end of Equation 28 are for conversion between English units and metric units and converting to kN.

4.251000

45.4'2⋅

⋅= mevhmi fAV (28)

mimf VV 3.0= (29) where: Vmi = available initial shear capacity that is consumed during the first half-cycle (monotonic) loading (kN)

Avh = net horizontal shear area of the infill panel (mm2) f’me = expected compressive strength of masonry prism (MPa)

Vmf = final shear capacity as a result of cyclic loading effects (kN) The net horizontal area used in Equations 28 and 29 is the product of the infill length and

thickness.

Results Comparison of Linear Model to Experimental Results

Table 5 shows the results of the comparison of the FEA displacement to the experimental

displacement at the first crack load level. The average percent difference from Table 5 is 2.4 percent with a standard deviation of 28.3 percent and a mean standard error of 5.4 percent. Since there are positive and negative percent differences, a better estimate would be to average the absolute value of the percent differences. Doing so results in an average percent difference of 20.9 percent with a standard deviation of 18.8 percent and a mean standard error of 3.6 percent. The standard deviation provides an indication of how much scatter is in the data, while the mean standard error gives an indication of how much of the scatter is due to the model itself. The mean standard error associated with the linear model indicates that the majority of the scatter is due to the experimental data rather than the FEA model.

The low average percent difference between the experimental displacement and the FEA displacement indicates that this FEA model is a good representation of the behavior of masonry infilled frames up to the first crack load. It should be noted that the percent difference for frames WC9L, WD3L, A4a, and C1 have not been used in the calculation of the average percent difference. The experimental displacement for these frames is 1.3 mm or less. For extremely low experimental displacements, even a small difference between the experimental displacement

18

and the FEA displacement results in a high percent difference. The actual difference between the experimental displacement and the analytical displacement for these frames is considerably better than the percent difference would indicate. In addition, the reported displacement for frames WC9L and WD3L is somewhat misleading. These are the displacements that occurred from the original frame location rather than from where the load returned back to zero along the second loading direction. In both of these cases the point at which the load returned to zero was still somewhat displaced from the starting position of the frame due to the loading in the first direction.

Several of the frames were loaded in both in-plane directions. The frames that were loaded in both directions have also been analytically modeled in both directions. The second in-plane direction is indicated by an “L” added to the FEA model identification number. The experimental displacement for the second in-plane direction for Richardson’s frames WD2 and WD4 is not available (i.e. WD2L and WD4L); however, the first major crack load was reported, thus the FEA model was used to analyze the second direction.

Figures 22 and 23 show the typical compressive stresses that develop in the FEA model. Figure 22 is a CMU infill surrounded by a steel frame (specifically FEA model MC1 which replicates McBride’s (1982) test 1), while Figure 23 is a RC infilled frame (specifically FEA model A4a which replicates Angel’s (1994) test specimen 4a). [These two frames, MC1 and A4a, will be used as representatives of the steel frames and the RC frames, respectively, in the following sets of figures.]

Figures 24 and 25 show the typical tensile stresses that develop in the infill. Figures 26 and 27 show the resulting von Mises stresses in the infill. The von Mises stress is an equivalent stress that takes into consideration all of the principal stresses. Typically, the von Mises stress is compared to the yield stress of the material. Figures 28 and 29 show the compressive strains while Figures 30 and 31 show the tensile strains. Figures 32 and 33 show the von Mises strains which develop in the infill material. Finally, Figures 34 and 35 show the force-displacement plots.

The compressive stress plots (Figures 22 and 23) clearly show the development of a diagonal strut within the masonry infill. As has been seen in many experimental studies, the diagonal strut carries a majority of the applied force with much of the masonry infill being near zero compressive stress. In fact the von Mises stress plots (Figures 26 and 27) show that a significant portion of the infill experiences very little stress under monotonic loading. The ends of the strut are in a state of bi-axial compression due to confinement by the column and the beam which has also been observed experimentally.

The strain plots (Figures 28 to 33) also show the diagonal strut that develops in the masonry infill. The compressive strains (Figures 28 and 29) and the von Mises strains (Figures 32 and 33) match the shape of their respective stress plots as expected from the basic relationship between stress and strain. The tensile strain plots (Figures 30 and 31), however, do not have the same form as the tensile stress plots (Figures 24 and 25). The reason for this apparent discrepancy is that the FEA plots are actually based on the principal stresses and principal strains. The compressive stress plots are the third principal stress, defined to be the most negative stress, while the tensile stress plots are the first principal stress, defined to be the most positive. The areas of high compressive stress will have high compressive strain as well as some tensile strain based on Poisson's ratio for the material. The areas of high tensile stress will have high tensile strain, but not necessarily compressive strain. So, the tensile strain plot will include the high tensile areas as well as some areas that are primarily in compression. Those areas taken

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together would give a tensile strain plot that appears to be a combination of the tensile stress plot and the compressive stress plot.

Finally, the force-displacement plots (Figures 34 and 35) are linear in nature. This matches the experimental force-displacement relationship up to the first crack load. After the first crack load the force-displacement relationship is typically no longer linear.

Comparison of Non-linear Model to Experimental Results

The non-linear FEA model has been compared to the experimental displacements at the first crack load and at the ultimate load. Table 6 shows the results of the comparison of the FEA displacement to the experimental displacement at the first crack load level. The average percent difference from Table 6 is -7.4 percent with a standard deviation of 31.0 percent and a mean standard error of 6.1 percent. The absolute value percent difference is 23.0 percent with a standard deviation of 21.5 percent and a mean standard error of 4.2 percent. As previously discussed, the non-linear model is somewhat dependent upon the results from the linear model in that it uses the maximum strain level from the linear model to determine which elements to “kill.” The average percent difference from the non-linear model at the first crack load is due to the use of the tri-linear stress strain curve rather than the constant modulus of elasticity utilized in the linear model.

Table 7 shows the results of the comparison of the FEA displacement to the experimental displacement at the ultimate load level. The average percent difference from Table 7 is 4.3 percent with a standard deviation of 48.8 percent and a mean standard error of 8.8 percent. When the absolute value of the percent differences is used the average percent difference is 37.9 percent with a standard deviation of 30.3 percent and a mean standard error of 5.4 percent. (Two of the FEA models, WD2L and HN1, did not converge to a final solution at the ultimate load level and thus are not included in the statistics listed here.) As with the linear model results, the mean standard errors associated with the non-linear model indicate that the majority of the scatter is due to the experimental data.

The representative stress and strain plots shown here are from the FEA model WC7. Figures 36 to 39 show the typical compressive stresses that develop in the FEA model at the different loading stages. Figure 36 shows the compressive stress that has developed in the infill at the first crack load. Figure 37 is the compressive stress after half of the difference between the first crack load and the ultimate load has been applied to the structure. This load has been termed the second crack load. Figure 38 shows the compressive stress at the ultimate load and Figure 39 is for the added 2 kN of loading beyond the ultimate load. The remaining sets of figures will follow this pattern of first crack load, second crack load, ultimate load, and 2 kN beyond ultimate load. Figures 40 to 43 show the typical tensile stresses that develop in the infill. Figures 44 to 47 show the resulting von Mises stresses in the infill. Figures 48 to 51 show the compressive strains while Figures 52 to 55 show the tensile strains. Figures 56 to 59 show the von Mises strains which develop in the infill material. Finally, Figures 60 to 92 show the force-displacement or P-delta plots for each of the frames investigated.

The low average percent difference between the experimental displacement and the FEA displacement at both the first crack load and the ultimate load indicates that this non-linear FEA model accurately replicates the behavior of masonry infilled frames. As was done with the linear model the percent difference for frames WC9L, WD3L, A4a, and C1 have not been used in the calculation of the average percent difference due to low experimental displacements. The difference between the experimental displacement and the FEA model displacement for frames

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A4a and C1 are less than 1 mm; however, the percent differences for these frames are negative 124 percent and negative 313 percent, respectively. The FEA model for frame HN1 did not converge to a solution at the first crack load level. The FEA model for frame WD2L converged at the ultimate load level, but failed to converge with the additional 2 kN of load.

The compressive stress plots (Figures 36 to 39) show the development of the diagonal strut within the masonry infill as the loading progresses. Initially the compressive stresses are virtually symmetrical; however, after each crack is modeled this symmetry begins to dissipate. This also indicates that this FEA model accurately represents the real world conditions within the masonry infill. Experimentally more damage is typically seen in the upper loaded corner of the masonry infilled frame. The results of the FEA model also show slightly higher stresses, and therefore more damage, in the upper loaded corner.

Comparison of the von Mises stress plots (Figures 44 to 47) with the compressive stress plots (Figures 36 to 39) shows that the diagonal strut continues to carry the majority of the applied load in compression and that a significant portion of the masonry infill experiences very little stress. This again matches well with experimental observations.

The same relationships between the stress and strain plots exist in the non-linear FEA model as did in the linear model. The compressive stress plots (Figures 36 to 39) and the compressive strain plots (Figures 48 to 51) as well as the von Misses stress plots (Figures 44 to 47) and the von Mises strain plots (Figures 56 to 59) have basically the same form. The tensile stress plots (Figures 40 to 43) and the tensile strain plots (Figures 52 to 55) are not the same form due to the tensile strain in areas of high compressive stress, as discussed previously in the linear model description.

One interesting phenomenon that can be seen in the von Mises strain plots (Figures 56 to 59) is the reduction of the width of the diagonal strut as cracking occurs. Figure 56 shows that prior to and immediately following the first major crack developing in the masonry infill the diagonal strut is wide and well defined. The strut width is still well defined in Figure 57; however, its width is slightly reduced. As loading continues and more cracks develop, the strut narrows even more and its effectiveness is reduced (Figure 58). Finally, Figure 59 shows that after the ultimate load the diagonal strut is essentially no longer active in resisting the applied lateral loads.

Figures 60 to 92 show the comparison of the experimental force-displacement plots to the FEA force-displacement plots for each frame. The experimental curve is blue and shows specific points on the force-displacement curve, while the force-displacement curve developed from the FEA model is smooth and magenta in color. Each of these figures is plotted to the same force and displacement scale to avoid skewing the results either for or against individual frames. The one exception to this is Figure 81 which shows the P-delta relationships for frame WD7. As can be see this particular frame was loaded well beyond the ultimate load level so its displacement scale is considerably longer than the other frames. Figure 87 which is for the HN1 model results shows only the experimental force-displacement curve since this particular model did not converge before reaching the first crack load.

Figures 76 and 80 show the analytical displacements plotted for models WD2L and WD4L. The researchers did not report experimental force-displacement relationships for these frames in the second loading direction. The experimental force-displacements plotted in these figures are the first loading direction force-displacements for frames WD2 and WD4. Theoretically, the strength and stiffness should be identical for each in-plane loading direction for any given masonry infilled frame. In reality there are slight differences due to the fact that

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loading in the first in-plane direction will result in minor changes. These changes could be in the form of small cracks in the infill or a reduction in adhesion between the infill and the frame.

The degradation of the strut width is also seen in the force-displacement plots (Figures 60 to 92). Figure 66 is the P-delta plot for frame WC7. The stiffness of the masonry infilled frame can be calculated as the slope of the P-delta plot. As Figure 66 shows the slope decreases after each crack occurs both experimentally and analytically. Based on the equivalent diagonal strut philosophy the width and effectiveness of the strut must be reduced in order for this to occur. This is especially important when considering to what load level the equivalent diagonal strut philosophy remains valid.

Overall the comparisons of the experimental force-displacement relationships to the FEA force-displacement relationships are excellent. All of the experimental force-displacement relationships show a significant increase in the displacement or a decrease in the slope (i.e. stiffness of the system) immediately following the first major crack in the infill. Thirty-one of the FEA P-delta plots show the same type behavior following the application of the first crack load. The FEA model that did not show either a large increase in displacement or a change in the stiffness of the system is model A4a. As can be seen in Figure 91 the experimental displacement did not have a large increase after the first crack, but there was a change in the stiffness of the system.

Nineteen of the FEA models exhibited a rapid increase after the application of the second crack load. Experimentally twelve of the frames had at least one additional crack between the first crack load and the ultimate load. Twenty of the analytical models showed a significant increase in displacement when loaded beyond the ultimate load while sixteen of the experimental frames showed the same behavior. It should be noted that many of the experimental frames were not loaded significantly beyond the ultimate load due to the desire to reuse the steel frame in subsequent testing (McBride 1982). The percentage of analytical displacements that are higher than the experimental displacements is roughly equal to the percentage of analytical displacements that are lower than the experimental displacements. While additional load steps might improve the displacement characteristics of some frames, it would likely be detrimental to the displacement characteristics of other frames.

Comparison of Stiffness Equations to Experimental Results

Tables 8, 9, and 10 are the comparisons of the existing stiffness techniques with the experimental data for CMU infill. Table 8 shows the comparison of the existing techniques to the experimental data for the steel frames, while Table 9 is the comparison between the existing techniques and the RC frames. Table 10 shows the combined results for all frames infilled with CMU. Table 8 is based on twenty-eight experimental tests; Table 9 is based on three experiments; and Table 10 combines the experimental data for a total of thirty-one experimental data points.

Each method estimates the width of the equivalent diagonal strut (See Figure 93). The second column is the average ratio of the predicted to experimental in-plane stiffness of the infill (“Average kpred / kexp ratio.”) The calculated stiffness, “kpred,” is based on a standard statics analysis using the area of the strut as the calculated width times the actual thickness of the infill. The experimental stiffness, “kexp,” is the stiffness at the first cracking load, which has been deemed to be a more consistent measure of the experimental stiffness. The tabulated value is the average ratio for the comparison to each experimental test. The ratio is the basis for the ordering of the table.

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Several analytical methods shown in Tables 8, 9, and 10 have multiple equations based

on the value of λ, which is the relative stiffness of the infill to the frame; the aspect ratio of the frame; and whether the infill is cracked or uncracked. Mainstone (1971) has an equation for λ between 4 and 5 and an equation for λ greater than 5 for both concrete masonry and brick masonry infilling. Decanni and Fantin (1986) have four equations; two for cracked infill and two for uncracked infill. Both sets of equations are split for λ less than 7.85 and λ greater than or equal to 7.85. Al-Chaar (2002) and Papia, et al. (2003) have equations for the aspect ratio greater than or equal to 1.5 and for the aspect ratio equal to 1.0. Both methods recommend interpolating if the actual aspect ratio is between 1.0 and 1.5.

Additional analytical methods have also been investigated but were not compared to the experimental data due to the unique features associated with these approaches. First, a method was developed by Jamal, et al. (1992) to be used for hinged frames, but is not applicable to the fixed frames of this experimental series. Bazan and Meli (1980) and Fiorato, Sozen, and Gamble (1970) developed equations that include the masonry shear modulus. While the shear modulus can be calculated using the elastic modulus and Poisson’s ratio, accurate results are difficult to achieve due to the excessive amount of estimation necessary for the associated variables

Some additional methods included in the presentation of existing equations are not shown in Tables 8, 9, and 10. The FEMA 356 (2000) method is not shown because it matches the equations given by Mainstone (1971). The method given by the New Zealand code (2004) matches the equation given by Paulay and Priestly (1992) and therefore is not listed in the tables.

The same set of existing in-plane stiffness methods is ranked at the top of Tables 8, 9, and 10. In fact Table 8 and Table 10 rank the methods in almost identical order. This is not unexpected since Table 10 is based on the same twenty-eight data points of Table 8 with an additional three tests. On average all of the existing equations over-predict the stiffness of the masonry infilled frame system at the first crack load. As with the FEA models, the mean standard errors associated with the better stiffness methods indicate that the majority of the scatter is due to the experimental data. The coefficient of variation (COV) given in the tables for each of the existing methods is the ratio of the standard deviation to the average expressed as a percentage. The COV is useful for comparing the amount of variation from one method to the next. A small COV indicates a method less scatter from the average for that method.

A review of the equations from each of the top methods in Table 10 shows several commonalities. Mainstone (1971), Bennett et al. (1996), Decanni and Fantin (1986) and MSJC Proposed (2007) equations include the relative stiffness factor λ in their equations for calculating the equivalent strut width. This factor is used to include the relative stiffness of the frame to the infill. El-Dakhakhni, et al (2003) also includes a measure of the frame stiffness with respect to the infill strength in his methodology. All of the top methods include the effect of the geometry and aspect ratio of the frame by including either the diagonal length of the equivalent strut or the angle of inclination of the strut. Five of the methods include the height of the infill. A majority of the equations above were developed by curve fitting techniques. These commonalities have been utilized in developing a new strut width equation and/or revising an existing equation.

Comparison of Strength Equations to Experimental Results

Table 11 shows the statistics for the strength comparisons at the first crack load. The

ratio is the predicted strength, Ppred, divided by the experimental strength, Pexp1. Table 12 shows

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the statistics for the strength comparisons at the reported ultimate load. A review of the experimental force-displacement curves (see Figures 60 to 92) shows that some of the frames might have had additional load capacity when loading was terminated; nevertheless, the load was reported as the ultimate load. Again the ratio is the predicted strength, Ppred, divided by the experimental strength at ultimate load, Pexp-ult. Tables 11 and 12 are based on the results of 30 steel frames and four RC frames infilled with CMU.

Each method estimates the horizontal strength of the masonry infilled frame system. Some methods were stated to be specifically for the first crack load or for the ultimate load and a particular failure mechanism. Since there is some question as to whether the true ultimate load was achieved for all of the frames, it was decided to compare each method to both the first crack load and to the reported ultimate load for completeness of the study. Tables 11 and 12 rank the methods in the same order; however, the ratios vary significantly between the two as would be expected.

A few of the existing methods (FEMA 306, 1999; Flanagan and Bennett, 1999; Mehrabi et al, 1996; Klingner and Bertero, 1976; and Flanagan, 1994) perform relatively well when compared to the experimental data; however, a majority of the methods significantly over-predict the strength of the masonry infilled frame system.

As with the stiffness methods, there were several strength methods investigated (listed in Appendix F) that were not compared to the experimental data. The reason for not comparing these methods to the experimental data is they require some material property not reported by the experimental studies, i.e. the tensile strength, the shear bond strength, or the shear strength of the infill.

Looking at the top nine strength methods between Tables 11 and 12 reveal the following similarities. All of these methods use the thickness of the infill in determining the strength. Six of the methods (Equations 20, 22, 23, 26, 28, and 29) use the infill compressive strength. Five of the methods use the length of the infill (Equations 21, 22, 25, 28, and 29). Three of the methods use a strut in the analysis (Equations 17, 20, and 26) and three of the methods use the drift of the frame (Equations 17, 20, and 25).

Proposed Code Recommendations for Stiffness and Strength

Several new equations have been developed and investigated in this study. These can be

categorized into two areas, revised equations and new equations based on regression analysis. Three equations were developed simply by modifying existing equations, while six new equations were developed by regressing stiffness on various infill parameters. Equations 30 to 38 are the new or revised equations that have been investigated during this study.

θλ cos25.0

=w (30)

θλ cos2.0

=w (31) 4.0)(05.0 −= hdw λ (32)

−1.15= h)0.25d( w λ (33) −1.1= d)0.35d( w λ (34)

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h700 w λ

= (35) −0.9)= d950( w λ (36)

−0.9= λ0.45 w (37)

λ45.0

=w (38)

where: w = width of equivalent strut (mm) λ = characteristic stiffness parameter (mm-1) θ = slope of infill diagonal to the horizontal h = height of frame (on centerlines of beams) (mm) d = diagonal length of infill (mm)

Equations 30 and 31 are modifications of the Proposed MSJC (2007) equation. Equation 32 is a modification of Equation 5 proposed by Mainstone (1971). Equations 33 to 38 are new regression models are based on the commonalities between the top existing methods. Equation 33 is from a regression of λh versus the calculated experimental strut width divided by the diagonal length (wexp/d). Equation 34 is from a regression of λd versus wexp/d. Equation 35 is from a regression of λh versus the calculated experimental strut width. Equation 36 is from a regression of λd versus the calculated experimental strut width. Equation 37 is from a regression of λ versus the calculated experimental strut width. Since λ-0.9 is very close to the inverse of λ, Equation 38 modifies Equation 37 as such.

Table 13 shows how these new and modified equations compare with the experimental results. The best existing equations are also shown for comparison purposes. All of the new equations, except for Equation 38, have a lower average ratio, standard deviation, and MSE than the existing equations. Statistically, this means that most of the new equations (Equations 30 to 37) are better, more consistent predictors of the equivalent diagonal strut width and thus the stiffness of the masonry infilled frame system. Equations 32, 33, and 34 have an average ratio of kpred to kexp close to unity.

Of these three equations, Equation 33 is the preferred equation for determining the width of the equivalent diagonal strut and thus the stiffness of the masonry infilled frame system. Not only does it include a majority of the parameters used in the best existing equations, it also slightly over-predicts the stiffness of the infilled frame system. This is preferred when considering how seismic loads are calculated. Typically, the higher the stiffness of structural system, the higher the seismic load will be for a given building mass. Use of Equation 33 helps to prevent an under-prediction of the system stiffness, and thus an under-prediction of the associated seismic load.

The commonalities previously discussed suggest that an equivalent diagonal strut analysis is appropriate for determining the strength of a masonry infilled frame system. The strength of a diagonal strut would be a function of the area of the strut which is easily determined from the width (calculated by Equation 33) and the thickness of the infill. The strength would also be a function of the compressive strength of the infill. In order to translate the diagonal strength of the strut to a horizontal load capacity the diagonal strength should be multiplied by cosine of the strut angle with respect to the horizontal. This is the concept used by FEMA 306 (1999) in Equation 26; however, Equation 26 significantly over-predicts the strength at the first crack load (see Table 11, FEMA 306 Strut). Equation 26 also slightly over-predicts the strength of the system at ultimate load (see Table 12, FEMA 306 Strut).

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In order to provide the foundation for performance based design, two strength equations

have been developed; one for the strength at the first crack load and one for the ultimate load. Equation 39 was developed as the strength of the masonry infilled frame system at the first crack load. Limiting the capacity to the first crack load will not only allow for residual capacity in the structural system, but should also limit the cost of repair after significant loading events. Equation 40 was developed as the ultimate strength of the masonry infilled frame system. This allows the engineer the option of using the residual capacity in the system beyond the first crack capacity.

θcos'6.0 wtfP mfc = (39) θcos'05.1 wtfP mult = (40)

where: Pfc = horizontal component of the diagonal strut capacity at first crack load (kN) Pult = horizontal component of the diagonal strut capacity at ultimate load (kN)

w = equivalent strut width per Equation 33 (mm) t = thickness of infill (mm) fm’ = compressive strength of masonry (MPa) θ = slope of infill diagonal to the horizontal (degrees)

Table 14 shows how Equation 39 compares to the experimental results, while Table 15 shows the results for Equation 40. The best existing strength method results are also shown in Tables 14 and 15 for comparison. As can be seen Equation 39 and 40 slightly under-predict the strength of the masonry infilled frame system. This is preferable in order to be conservative in the design of the system. Since this study used data from unreinforced, ungrouted masonry experiments, it is important that these stiffness and strength equations (Equations 33, 39, and 40) be limited to ungrouted walls.

Conclusions and Recommendations The linear model is a good representation of the behavior of masonry infilled frames up

to the first crack load. The FEA displacements closely match the displacements determined experimentally. The FEA stresses and strains are also a close approximation to the stresses and strains that have been observed experimentally. Since the experimental force-displacement relationship is non-linear beyond the first crack load, the results of this linear model are helpful, but incomplete.

The non-linear model provides a good representation of the behavior of masonry infilled frames up to the ultimate load. The FEA force-displacement relationships accurately replicate the experimentally determined relationships. The FEA stresses and strains are similar to experimental observations.

The proposed stiffness and strength equations (Equations 33, 39, and 40) provide an accurate, yet conservative, estimation of the behavior and capacity of frames infilled with CMU. In addition, they are theoretically sound and simple to use. As such, use of these equations should allow engineers to take advantage of the easy and economical construction of infills, as well as their strength and ductility in resisting lateral loads.

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