ANALYTICAL MODEL FOR LATERAL DEFLECTION IN COLD-FORMED STEEL FRAMED SHEAR WALLS WITH STEEL SHEATHING. Mohamad Yousof

Transcription

1 ANALYTICAL MODEL FOR LATERAL DEFLECTION IN COLD-FORMED STEEL FRAMED SHEAR WALLS WITH STEEL SHEATHING Mohamad Yousof Thesis Prepared for the Degree of MASTER OF SCIENCE UNIVERSITY OF NORTH TEXAS December 214 APPROVED: Cheng Yu, Major Professor Michael Shenoda, Committee Member Huseyin Bostanci, Committee Member Enrique Barbieri, Chair of the Department of Engineering Technology Costas Tsatsoulis, Dean of the College of Engineering Mark Wardell, Dean of the Toulouse Graduate School

2 Yousof, Mohamad. Analytical Model for Lateral Deflection in Cold-Formed Steel Framed Shear Walls with Steel Sheathing. Master of Science (Engineering Systems-Construction Management), December 214, 115 pp., 17 tables, 38 figures, references, 7 titles. An analytical model for lateral deflection in cold-formed steel shear walls sheathed with steel is developed in this research. The model is based on the four factors: fastener displacement, steel sheet deformation, and hold-down deformation, which are from the effective strip concept and a complexity factor, which accounts for the additional influential factors not considered in the previous three terms. The model uses design equations based on the actual material and mechanical properties of the shear wall. Furthermore, the model accounts for aggressive and conservative designers by predicting deflection at different shear strength degrees.

3 Copyright 214 by Mohamad Yousof ii

4 ACKNOWLEDGMENT First, I would to thank Dr. Cheng Yu, my mentor, for the time, effort, patience, advice, and the financial support and shared knowledge he provided for this thesis. Also, I would like to thank my committee members, Dr. Michael Shenoda and Dr. Huseyin Bostanci, for their time and effort. I would also like express great appreciation and to thank my father, Hamzah Yousof, for his care, motivation, and ongoing support. Finally, I would like to thank my colleagues in the research team: Nathan, Mahsa, Emanuel, and Ravi. Special recognition goes to Martin for his help in conducting over 2 tests in a short period of time. iii

5 TABLE OF CONTENTS ACKNOWLEDGEMENT... iii LIST OF TABLES... vi LIST OF FIGURES... vii CHAPTER 1 INTRODUCTION...1 CHAPTER 2 RESEARCH OBJECTIVES...3 Page 2.1 Thesis Statement Research Objectives...3 CHAPTER 3 EXISTING DESIGN MODEL Deflection Design Method in AISI Analytical Model Effective Strip Model...9 CHAPTER 4 PROPOSED ANALYTICAL MODEL AND DESIGN EQUATIONS FOR LATERAL DEFLECTION Fastener Displacement by Connection Strength Tests Effective Strip Model Verification Deformation in Steel Sheathing Deformation in Hold-Down/Anchorage Analytical Model for Lateral Deflection Complexity Factor Summary...45 CHAPTER 5 CONCLUSIONS AND RECCOMMENDATIONS Connection Strength Tests Proposed Analytical Model...46 iv

6 APPENDIX A CALCULATION EXAMPLE AND RESULTS FOR EFFECTIVE STRIP MODEL VERIFICATION BY CONNECTION STRENGTH TESTS...48 APPENDIX B DESIGN EXAMPLE AND RESULTS FOR LATERAL DEFLECTION IN CFS-SW-SS BY PROPOSED MODEL...52 APPENDIX C DATA SHEETS OF CONNECTION STRENGTH TESTS...66 REFERENCES v

7 LIST OF TABLES Page Table 3.1 Comparison of lateral deflection results by equation Table 3.2 Comparison of nominal shear strength results...15 Table 4.1 Test matrix for connection strength tests...18 Table 4.2 Material properties for connection strength tests specimens...19 Table 4.3 Comparison of nominal shear strength results by connection strength tests...22 Table 4.4 Manufacturer s hold-down specifications (C-CFS1 21) Table 4.5 Shear wall tests...27 Table 4.6 Comparison of lateral deflection...28 Table 4.7 Comparison of lateral deflection results by the proposed model...3 Table 4.8 Summary list of scaling and exponent factors...37 Table 4.9 Lateral deflection comparison by proposed model with design equations...39 Table 4.1 Statistical analysis for proposed model...4 Table A.1 Comparison of nominal shear strength results, aspect ratio = Table A.2 Comparison of nominal shear strength results, aspect ratio = Table B.1 Table B.2 Table B.3 Lateral deflection results for shear walls by proposed model, aspect ratio = Lateral deflection results for shear walls by proposed model, aspect ratio = Lateral deflection results for shear walls by proposed model, aspect ratio = vi

8 LIST OF FIGURES Page Figure 1.1 Typical CFS light framed shear wall with sheathing...2 Figure 3.1 Shear wall label definition...8 Figure 3.2 Lateral deflections ratio versus λ...9 Figure 3.3 Diagonal tension field in shear walls...1 Figure 3.4 Connection failure modes in shear walls...1 Figure 3.5 Effective strip in steel sheet sheathing...11 Figure 3.6 Equilibrium of nominal tension force in sheathing...12 Figure 4.1 Fastener displacement in the effective strip...17 Figure 4.2 Connection strength test setup...18 Figure 4.3 Dominant failure mode in connection strength tests...2 Figure 4.4 Connection strength test label...2 Figure 4.5 Diagonal tension field in shear walls...23 Figure 4.6 Reaction forces of shear wall on hold-downs...25 Figure 4.7 Shear wall label definition for tests used...27 Figure 4.8 Lateral deflection difference ratio verses λ for proposed model...29 Figure 4.9 Lateral deflection ratio versus λ at peak load...3 Figure 4.1 Lateral deflection difference ratio versus λ, V/Vn= Figure 4.11 Lateral deflection difference ratio versus λ, V/Vn= Figure 4.12 Lateral deflection difference ratio versus λ, V/Vn= Figure 4.13 Lateral deflection difference ratio versus λ, V/Vn= Figure 4.14 Lateral deflection difference ratio versus λ, V/Vn= vii

9 Figure 4.15 Lateral deflection difference ratio versus λ, V/Vn= Figure 4.16 Lateral deflection difference ratio versus λ, V/Vn= Figure 4.17 Lateral deflection difference ratio versus λ, V/Vn= Figure 4.18 Lateral deflection difference ratio versus λ, V/Vn= Figure 4.19 Lateral deflection difference ratio versus λ, V/Vn= Figure 4.2 Scaling factor verses shear strength degree...37 Figure 4.21 Exponent factor verses shear strength degree...38 Figure 4.22 Lateral deflection difference versus λ using Eq. 5.1, V/Vn = Figure 4.23 Lateral deflection difference versus λ using Eq. 5.1, V/Vn = Figure 4.24 Lateral deflection difference versus λ using Eq. 5.1, V/Vn = Figure 4.25 Lateral deflection difference versus λ using Eq. 5.1, V/Vn = Figure 4.26 Lateral deflection difference versus λ using Eq. 5.1, V/Vn = Figure 4.27 Lateral deflection difference versus λ using Eq. 5.1, V/Vn = Figure 4.28 Lateral deflection difference versus λ using Eq. 5.1, V/Vn = Figure 4.29 Lateral deflection difference versus λ using Eq. 5.1, V/Vn = Figure 4.3 Lateral deflection difference versus λ using Eq. 5.1, V/Vn = Figure 4.31 Lateral deflection difference versus λ using Eq. 5.1, V/Vn= viii

10 CHAPTER 1 INTRODUCTION Cold-formed steel (CFS) has quickly merged into today s industry due to its exclusive characteristics such as durability, strength, economics, and weight. Extensive research on CFS applications has been conducted for the past few decades on how to implement its advantages into framing applications, mostly in low and mid-rise buildings. Mainly, the objective of research previously conducted was to analyze the factors affecting CFS Frames strength and economics. CFS framed shear walls with shear sheathing are a widely accepted option in low and mid-rise building given its endurable performance in resisting in-plane lateral loads which normally occur from wind and earth-quake loads. Typically, a light framed shear wall will transfer loads in the plane of the wall through the shear sheathing which is attached mechanically throughout the frame. When designing CFS frames with sheathing, one will need to take into consideration multiple factors. Some of these factors include: the design loads (wind, seismic), deflection of frame, the thickness of the studs/frame members, the connection type(s), the sheathing material, and the configuration of wall (wall could have bracing, reinforcing members, or openings). All of these factors affect the performance strength and deflection of the shear wall. Figure 1.1 shows a typical CFS light framed shear wall system with shear sheathing. The frame is 8 ft. in height and 4 ft. in width. It contains two vertical boundary members or studs (end studs), two horizontal members or tracks (bottom and top), and an interior member in its center. Sheathing, which can be steel or wood-based panels, is typically connected to the frame by injecting screws mechanically through the sheathing and frame members. 1

11 Figure 1.1 Typical CFS light framed shear wall with sheathing. (Yanagi and Yu 214) The International Building Code (IBC 26) requires that the design of light framed CFS shear walls with steel sheathing be in accordance with the American Iron and Steel Institute (AISI) Standard for CFS Steel Framing Lateral Design (AISI S213, 27). AISI standard recognizes two basic types of CFS shear walls. Type I is a fully sheathed shear wall which resists in-plane forces with hold-downs at the end of each wall segment, and where details for the force transfer around the openings is provided if the wall has openings. Type II shear wall contains several wall segments which resist in-plane forces, using wood-based sheathing or steel sheathing that contains openings between the walls segments and hold-downs only at the ends of the walls. 2

12 CHAPTER 2 RESEARCH OBJECTIVES 2.1 Thesis Statement When Cold-formed steel shear walls (CFS-SW) experience lateral loading, lateral deflection occurs and this is a critical aspect as excessive lateral deflection can lead to undesired cracks within the shear wall assembly or in the finishing material, and possible collapse. Lateral deflection that results from lateral loading on the shear wall is a significant concern that needs to be addressed. The development of an analytical model and closed form design equations to predict the lateral deflection of CFS-SW with steel sheet sheathing (CFS-SW-SS) without the need to perform full-scale shear wall test will be a substantial step in widening the design options designers can choose from. Therefore, the objective of this research is to develop an analytical model for lateral deflection in CFS-SW-SS undergoing lateral loading. 2.2 Research Objectives The primary objective of this research is to develop an analytical model to predict the lateral deflection of CFS-SW-SS when undergoing lateral loading based on the effective strip model developed in Yanagi and Yu (214). Experimental tests conducted by Dr. Cheng Yu and his research team on shear wall specimens and their corresponding results for shear strength and lateral deflection reported in Yu (21) and Yu and Chen (211) were used to develop and verify the analytical model. The specific objectives of this research include: i. Studying and integrating shear wall test data reported in Yu (21) and Yu and Chen (211) to obtain relevant information to determine factors which contribute to CFS-SW- SS lateral deflection. 3

13 ii. Developing an analytical model for lateral deflection based of the effective strip model developed in Yanagi and Yu (214) to predict the lateral deflection of CFS-SW-SS undergoing lateral loading. iii. Establishing design equations based on the analytical model that can estimate the lateral deflection of CFS-SW-SS with different framing conditions. 4

14 CHAPTER 3 EXISTING DESIGN MODEL Literature Review An existing design deflection model developed by Serrette (26) to predict lateral deflection of Cold-formed steel shear walls (CFS-SW) is discussed in this section. The deflection results generated by the model are compared to existing data for CFS-SW with steel sheathing (CFS-SW-SS) tests reported in Yu (21) and Yu and Chen (211). Furthermore, the effective strip model, a model used to predict the nominal shear strength of CFS-SW-SS in Yanagi and Yu (214) is discussed. 3.1 Deflection Design Method in AISI The American Iron and Steel Institute Steel Framing Standards-Lateral Design (AISI S213, 27) includes a design deflection model, developed by Serrette (26), shown under section C2.1.1 Design Deflection, which estimates defection of a blocked wood structural panel or a steel sheet shear wall fastened throughout the frame. The model combines four different empirical factors/effects which contribute to the total lateral deflection. The first factor estimates the linear elastic cantilever bending (boundary member contribution), the second factor estimates the linear elastic sheathing shear, the third factor estimates the contribution of the overall nonlinear effects, and the fourth factor estimates the lateral contribution resulting from anchorage/hold-down deformation. The four factors and their corresponding terms are: 2vvh3 ssss = bbbb + ssss + nnnnnn + hdd vvh 5 δδ = + ωω 3EE ss AA CC bb 1ωω 2 + ωω 4 ρρρρtt ssheeeeeehiiiiii 1 ωω 2 ωω 3 ωω 4 ( vv ββ )2 + h δδ bb vv (3.1) where: ssss = total design deflection (in.); 5

15 bbbb = linear elastic cantilever bending (in.); ssss = linear elastic sheathing shear (in.); nnnnnn = overall non-linear effects (in.); hdd = anchorage/hold-down deformation (in.); νν = shear demand (V/b), in pounds per linear inch (N/mm); h = wall height, in inches (mm); EE ss = modulus of elasticity of steel = 29,5, psi (23, MPa); AA CC = gross cross-sectional area of chord member, in square inches (mm 2 ); b = width of shear wall, in inches (mm); s = maximum fastener spacing at panel edges, in inches (mm); tt ssheeeeeehiiiiii = nominal panel thickness, in inches (mm); tt ssssssss = framing designation thickness, in inches (mm); ρρ = 1.85 for plywood and 1.5 for OSB; =.75(t sheathing /.18) for sheet steel (for t sheathing in inches); =.75(t sheathing /.457) for sheet steel (for t sheathing in mm); V = total lateral load applied to the shear wall, in pounds (N); β = 67.5 for plywood and 55 for OSB for U.S. Customary (lb/in 1.5 ); = 2.35 for plywood and 1.91 for OSB for SI units (N/mm 1.5 ); = 41.67(t sheathing /.18) for sheet steel (for t sheathing in inches) (lb/in 1.5 ); = 1.45(t sheathing /.457) for sheet steel (for t sheathing in mm) (N/mm 1.5 ); δδ vv = vertical deformation of anchorage / attachment details, in inches (mm); δδ = calculated deflection, in inches (mm); 6

16 ωω 1 = s/6 (for s in inches) and s/152.4(for s in mm); ωω 2 =.33/ t stud (for t stud in inches) and.838/ t stud (for t stud in mm) ωω 3 = h/bb 2 ; ωω 4 = 1 for wood structural panels; = 33 F y (for F y in ksi); = F y (for F y in MPa) for sheet steel. The four terms used in equation 3.1 were based on regression and interpolation analysis of the reversed cyclic test data used in the development of the cold-formed steel shear wall design values. The ρρ term in the linear elastic sheathing shear factor encompasses differences in the response of walls with similar framing, fasteners and fastener schedules, but different sheathing material. In order to verify the applicability of the model for CFS-SW-SS, lateral deflection results were calculated using equation 3.1 and compared with experimental lateral deflection results reported in Yu (21) and Yu and Chen (211). The results are presented in Table 3.1. The first column to the left lists the label of the shear wall configuration defined in Figure 3.1. The second column lists the aspect ratio of the shear wall, the third column lists the lateral design deflection results by equation 3.1, and the fourth column lists the experimental lateral deflection results reported in Yu (21) and Yu and Chen (211). 7

17 Figure 3.1 Shear wall label definition. Table 3.1 Comparison of lateral deflection results by equation 3.1. Shear Wall Configuration Aspect Ratio Deflection by Eq. 3.1 (in.) Deflection in Yu (21) and Yu and Chen (211) (in.) 2x8x33x x8x43x x8x33x x8x43x x8x43x x8x54x As illustrated in Table 3.1, equation 3.1 yields results that are considerably less than the experimental lateral deflection for CFS-SW-SS. This is because the model was developed based on a rigid/thick sheathing assumption in which the panel would yield relatively small lateral deflection due to panel deformation as the frame of the shear wall (studs/vertical members) bends, while the extensive experiments have revealed that the CFS-SW-SS resist lateral loads by a diagonal tension field action in the sheathing (explained in section 3.1.2). Therefore, the lateral deflection due to the sheathing is primarily contributed by the diagonal tensile elongation in the sheathing. Figure 3.2 graphically illustrates the under-estimated results of lateral deflection using equation 3.1. The design formula λ is a ratio based on actual measurements of material thickness and mechanical properties, which was developed for the effective strip method in Yanagi and Yu (214). 8

18 Δexp Δsw λ Figure 3.2 Lateral deflections ratio versus λ. 3.2 Analytical Model - Effective Strip Model When experiential laboratory tests on CFS-SW-SS were conducted, a diagonal tension field extending from corner to corner was observed. This is shown in Figure 3.3. It was also observed that the dominant failure mechanism in the CFS-SW-SS wall specimens tested was fastener connection failure within the diagonal tension field, as shown in Figure 3.4. This indicated that the steel sheathing did not equally contribute/transfer the shear resistance across the width of the entire shear wall. Instead, the diagonal tension field endured most of the tension force that occurred in the system, and this observation led Yanagi and Yu (214) to create an analytical model to predict the nominal shear strength called the effective strip model. The model assumed that in the diagonal tension field created when undergoing lateral loading, an effective strip of particular width is providing the shear resistance in the system. This is illustrated in Figure

19 AR=4 AR=2 AR=1.33 Figure 3.3 Diagonal tension field in shear walls. (Yanagi and Yu 214) Figure 3.4 Connection failure modes in shear walls. (Yanagi and Yu 214) 1

20 Figure 3.5 Effective strip in steel sheet sheathing. (Yanagi and Yu 214) In Figure 3.5, WW ee is the width of the effective strip of the sheathing, T is the resulting tension force in the effective strip in the sheathing occurring when VV aa, the lateral load, is applied, α is the angle at which the tension force acts, and h and W are the wall s height and width respectively. By analyzing the forces in Figure 3.5, the applied lateral load VV aa can be expressed as: VV aa = TT cos αα (3.2) Equation 3.2 indicates that the applied lateral load is directly related to the tension force experienced in the steel sheathing. Therefore, the maximum force obtained from the shear wall system is limited by the maximum tension force in the sheathing. The maximum tension force in the sheathing is then limited to the capacity of the sheathing-to-framing connections at both ends of the effective strip and the sheathing material yield strength of the effective strip. The nominal shear strength of the fastener connection used in the sheathing-to-framing connection is limited 11

21 to three modes of failure in consideration with the North American Specification for the Design of Cold-Formed Steel Members (AISI S1, 27) and those are:. i) Connection shear limited by tilting and bearing. ii) Connection shear limited by end distance measured in line of force from the center of a standard hole to the nearest end of the connected parts. iii) Shear failure in fastener (screws). With this in mind, Yanagi and Yu (214) were able to develop an experimental model which predicted the nominal shear strength of CFS-SW-SS based on the three modes of failure in sheathing-to-framing connection mentioned above and the sheathing material yield capacity on the effective strip. The sheathing-to-framing connection shear strengths can determined based on the smallest value obtained by following E4.3.1, E4.3.2, and E4.3.3 of AISI S1 (27). Figure 3.6 illustrates the behavior of the sheathing-to-framing fasteners. Figure 3.6 Equilibrium of nominal tension force in sheathing. (Yanagi and Yu 214) 12

22 Equation 3.3 illustrates the effective strip model equation developed by Yanagi and Yu (214). One has to determine the sum of the nominal shear capacity of the individual connections between sheathing-to-track, sheathing-to-stud, and sheathing-to-track-stud at wall corners, and determine the sheathing yield capacity. The minimal value of the two controls and is multiplied by the determined reduction factor =.79, provided by LRFD design shear strength of the shear walls. where: VV nn = mmmmmmmmmmmmmm WW ee PP 2ss ssssssss nnnn,tt + WW ee PP 2ss cccccccc nnnn,ss + PP nnnn,tt&ss cccccccc, WW ee tt ssh FF yy cccccccc (3.3) VV nn = nominal shear strength in shear wall (lb.); WW ee = effective strip width of sheathing (in.); PP nnnn,tt = nominal shear strength of fastener at sheathing-to-track connection (lb.); PP nnnn,ss = nominal shear strength of sheathing-to-stud connection (lb.); PP nnnn,tt&ss = nominal shear strength of sheathing-to-track-stud connection at wall corners (lb.); FF yy = yield capacity of sheathing material (ksi); tt ssh = sheathing material thickness (in.); s = perimeter screw spacing (in.); α = angle at which the tension force acts (deg.). The design formulas for the effective strip width were based on actual measurements of material thickness and mechanical properties. The proposed formula for the effective strip width is as listed. WW ee = WW mmmmmm, iiii λλ.819 ρρww mmmmmm, iiii λλ >.819 (3.4) ρρ = 1.55(λλ.8).12 λλ.12 (3.5) 13

23 λλ = αα 1αα 2 ββ 1 ββ 2 ββ 3 2 aa (3.6) where: WW ee = effective strip width of sheathing (in.); WW mmmmmm = maximum width of effective strip (in.); = w/sinα; a = aspect ratio of shear wall; = h ww ; αα 1 = FF uuuuh /45; αα 2 = FF uuuuh /45; ββ 1 = tt uuuuh /.18; ββ 2 = tt uuuuuuuu /.18; ββ 3 = s/6; s = perimeter screw spacing (in.); FF uuuuh = tensile strength of steel sheet sheathing (ksi); FF uuuuuuuu = controlling tensile strength of framing member (ksi); tt uuuuh = thickness of steel sheet sheathing (in.); tt uuuuuuuu = smaller thicknesses of track and stud (in.). Table 3.2 shows the comparison between the published nominal shear strength values of AISI S213 (27) wind and seismic shear strengths and the nominal shear strength values determined using the effective strip model. The predicted nominal shear strength values are almost equivalent or within an accepted range compared to the published values. The approach 14

24 towards developing an analytical model for lateral deflection is entirely based on factors involved in the effective strip model theory. Table 3.2 Comparison of nominal shear strength results. Shear Wall Configuration AISI S213 (27) Table C2.1-1 (plf) AISI S213 (27) Table C2.1-3 (plf) Predicted Vn (plf) 2:1x33x :1x43x :1x43x :1x43x :1x33x :1x33x :1x33x :1x33x

25 CHAPTER 4 PROPOSED ANALYTICAL MODEL AND DESIGN EQUATIONS FOR LATERAL DEFLECTION Effective Strip Model for Lateral Deflection Based on the effective strip model for lateral strength, three mechanical factors: fastener displacement, sheathing deformation, and hold-down deformation contribute to the total lateral deflection of the col-formed steel shear walls with steel sheathing (CFS-SW-SS). As mentioned previously, the diagonal tension field of a certain width (effective strip) provides most of the resistance for lateral loading. Therefore, fasteners in the effective strip experience relative displacement, and this yields (allows) lateral deflection of the shear wall. Also, deformation (elongation) in the highly tensioned effective strip occurs and contributes towards lateral deflection of the shear wall. In addition, when the lateral load is applied, the shear wall tends to tilt in the direction of the load. Vertical and horizontal movement is prevented by the hold-downs used as anchorage in shear wall systems. This results in hold-down deformation (elongation) which relatively contributes to the total lateral deflection of the wall. Therefore, hold-down deformation (elongation) should be considered in the development of the analytical model. 4.1 Fastener Displacement by Connection Strength Tests Part of the total lateral deflection CFS-SW-SS experience results from the displacement of the fasteners at the sheathing-to-framing connections within the effective strip when the lateral load is applied. This is illustrated in Figure 4.1. The extent of contribution which yields from the displacement of fasteners into the lateral deflection of the shear wall system was analyzed in order to develop an improved analytical model for lateral deflection. 16

26 Figure 4.1 Fastener displacement in the effective strip. Figure 4.2 illustrates the setup made to conduct connection strength tests for the analysis. Basically, a 4 in. 4 in. steel sheet was connected by a single fastener (screw) to the flange of a 6 in. in length CFS track member. A steel plate was placed on the web of the track member to fix it to the test bed throughout by four C clamps. This was done to prevent any movement in the setup while the tension force was pulling the steel sheet (sheathing) upward. Four bolts having 5/8 in. in diameter were used to connect the steel sheet with the ascending pulling arm/connector (thick bolts were used to prevent excessive elongation in the steel sheet/sheathing), and a piece of wood was inserted in the gap between the sheathing and the pulling arm to prevent horizontal movement of the steel sheet. Table 4.1 shows the test matrix of 96 connection strength tests of different specifications conducted in this study. The peak load and displacement at peak load were reported as illustrated. The material properties of the specimens used in the connection strength tests are listed in Table

27 Figure 4.2 Connection strength test setup. Table 4.1 Test matrix for connection strength tests. Fastener No.8 No.1 No.12 Steel Sheet (Sheathing)(mil) Pu(lb.) Δ(in.) Pu(lb.) Δ(in.) Pu(lb.) Δ(in.) Pu(lb.) Δ(in.) Pu(lb.) Δ(in.) Pu(lb.) Δ(in.) Pu(lb.) Δ(in.) Pu(lb.) Δ(in.) Framing Member(mil) 18

28 Table 4.2 Material properties for connection strength tests specimens. Member Sheathing Framing Member Fastener Specification Measured Properties Thickness (in.) Fy (ksi) Fu (ksi) 18 mil mil mil mil mil mil mil mil Type Washer Dia. (in.) Dia. (in.) No No No It is important to keep in mind that the material properties of each specimen used in the connection strength tests affects the load and displacement, and that thickness is not the single factor which can vary load and displacement. Values reported in Table 4.1 are only for specimens used in this experiment. In general, sheathing build-up and tear under the fastener was the dominant failure mode observed in most connection strength tests. Figure 4.3 illustrates such a failure mode. Also, in tests 68x43x8 1 and 2, shear failure in No.8 self-drilling screws was observed. Appendix C contains the specifications of the each material used, corresponding plots, and test results for all connection strength tests conducted in this research. The definition used in the connection tests is illustrated in Figure 4.4. To avoid confusion, please disregard the label shown in Figure 4.3 and use the definition shown in Figure

29 Figure 4.3 Dominant failure mode in connection strength tests. Figure 4.4 Connection strength test label. To maintain consistency with the effective strip model, the shear load each fastener in the effective strip experiences was determined and the corresponding displacement at that load was used as the fastener displacement in the analytical model for lateral deflection. In addition, since the diagonal tension field in the effective strip in CFS-SW-SS extends from corner to corner, fasteners on both corners experience an inward (in the direction of the tension pulling force) pulling tension force and therefore the displacement of fasteners should be multiplied by 2. When the displacement of fasteners is projected to lateral deflection of the shear wall, the model for lateral deflection can be expressed as: llllll = 2 ff cos αα (4.1) where: llllll = lateral deflection of shear wall (in.); 2

30 ff = displacement of fasteners (in.), average displacement of test 1 and test 2 from Table Effective Strip Model Verification As mentioned earlier, the maximum resisting force obtained from the shear wall system is limited by the maximum tension force in the sheathing. The maximum tension force in the sheathing is limited to the capacity of the sheathing-to-framing connections at both ends of the effective strip and the sheathing material yield strength of the effective strip. Therefore, the capacity of the sheathing-to-framing connections limits the nominal shear strength of CFS-SW- SS. Based on the effective strip model, the number of fasteners on the track and the stud in the effective strip can be determined by equations 4.2 and 4.3. To verify the effective strip model, the number of fasteners on the track and stud were added and then multiplied by the peak load obtained from connection strength tests. The result should match the nominal shear strength of the CFS-SW-SS or fall within an acceptable range. A calculation example is provided in Appendix A along with all the calculated results for each shear wall. Note that equations 4.2 and 4.3 do not account for the fastener connecting the stud, track, and sheathing at the corner. Therefore, 1 should be added to the total number of fasteners in the effective strip. nn tt = nn ss = WW ee 2ss ssssssss WW ee 2ss cccccccc (4.2) (4.3) where: nn tt = total number of sheathing-to-framing fasteners on track; nn ss = total number of sheathing-to-framing fasteners on stud. 21

31 Table 4.3 lists several CFS-SW-SS experimental tests conducted at the University of North Texas along with their reported experimental nominal shear strength from Yu (21) and Yu and Chen (211). When compared with the shear strength obtained by connection strength tests, it was concluded that the difference was minimal, the average test to predicted ratio was for all the specimens was 1.4, and the effective strip model was verified. Table 4.3 Comparison of nominal shear strength results by connection strength tests. Shear Wall Configuration Experimental Shear Strength in Yu (21) and Yu and Chen (211) (plf), Vtest Shear Strength by Connection Strength Tests (plf),vn Vtest/Vn 2x8x33x27-6/12-M x8x33x27-2/12-C x8x33x27-6/12-M x8x43x33-2/12-C x8x43x27-2/12-M1-D x8x54x33-2/12-C2-B M: Monotonic C: Cyclic Average Deformation in Steel Sheathing As illustrated earlier in Figure 3.3, shown again in this section, a bubble-like form is visible in the effective strip of the steel sheathing as a result of deformation (elongation) due to resisting the applied lateral load. This observation led to the incorporation of the deformation of the sheathing into the total lateral deflection of CFS-SW-SS as it contributes into lateral deflection of the shear wall. 22

32 AR=4 AR=2 AR=1.33 Figure 4.5 Diagonal tension field in shear walls. (Yanagi and Yu 214) To determine the diagonal deformation of the steel sheathing, the effective strip s width and length were calculated. The effective strip s length was calculated from corner to corner. The tension force in the steel sheathing was calculated by directly relating the applied lateral load to the tension force experienced in the steel sheathing as shown earlier in equation 3.4. The applied lateral load was obtained from reported values in Yu (21) and Yu and Chen (211). Equation 4.4 was used to determine the diagonal deformation of the sheathing. Δ dddd = TTLL ee WW ee tt ssh EE (4.4) where: dddd = diagonal deformation of steel sheathing (in.); T = diagonal tension force in steel sheathing (lb.) (determined by equation 3.4); LL ee = effective strip length (in.); = h 2 + ww 2 ; h = height of shear wall (in.); 23

33 ww = width of shear wall (in); The diagonal deformation of steel sheathing is projected towards the lateral deflection of the shear wall in the same manner of displacement of the fasteners. Now, the model for lateral deflection can be expressed as: llllll = 2 ff + Δ dddd cos αα (4.5) where: dddd = diagonal deformation of steel sheathing (in.). 4.4 Deformation in Hold-Down/Anchorage Hold-downs anchored to a steel box beam (connected to the fixed frame) by steel bolts were used to restrain CFS shear wall movement while undergoing loading. Figure 4.5 illustrates how the hold-downs resist the tension and compression forces created by the applied lateral load. The deformation (extension) in the hold-down opposing the tension force (left hold-down) occurs in the bolt used for hold-down anchorage and the hold-down itself. This deformation contributes to the lateral deflection of the shear wall and was addressed to develop an improved analytical model for lateral deflection. Properties to determine the value of deformation in the tensioned hold-down of the CFS-SW-SS tests conducted in Yu (21) and Yu and Chen (211) were provided by the manufacturer and are listed in Table 4.4. Values for the deflection at LRFD load in Table 4.4 include the deformation of both the hold-down and the anchorage bolt from the manufacturer s catalog C-CFS1 (21). 24

34 Figure 4.6 Reaction forces of shear wall on hold-downs. Table 4.4 Manufacturer s hold-down specifications (C-CFS1 21). Hold- Down Model H (in.) Fasteners Found. Anchor Diameter (in.) Stud Fasteners S/HD1S / #14 LRFD Deflection at Tension Load (lb.) LRFD Load (in.) 11, , , Under shear wall lateral loading, the hold-down resisting tension experiences an upward tension force which can be expressed as: PP hdd = VV aa ww aa (4.6) where: PP hdd = vertical tension force experienced by hold-down (lb.); 25

35 VV aa = applied lateral Load (plf); Using the properties provided by the hold-down manufacturer, the vertical deformation of the hold-down with the anchorage bolt can be expressed as: where: vvvv = PP hdd tt PP tt (4.7) vvvv = vertical deformation of the hold-down with the anchorage bolt (in.); tt = deflection at corresponding LRFD load (in.) from Table 4.4; PP tt = tension load (lb.) from Table 4.4. The contribution of the vertical deformation of the hold-down with the anchorage bolt towards the lateral deflection of the shear wall can be expressed as: where: aaaaaah = vvvv aa (4.8) aaaaaah = hold-down/anchorage deformation towards lateral deflection (in.). Now, the model for lateral deflection can be expressed as illustrated equation 4.9. Equation 4.9 is the analytical model developed which summarizes all the three factors initially considered in this research that contribute to lateral deflection of CFS-SW-SS based on the effective strip model. llllll = 2 ff + Δ dddd cos αα + aaaaaah (4.9) 4.5 Analytical Model for Lateral Deflection Table 4.5 lists the shear wall tests included in the development of an analytical model for lateral deflection and in the development of the effective strip model. When lateral deflection results from equation 4.9 were compared with the experimental lateral deflection results reported 26

36 in Yu (21) and Yu and Chen (211), it was observed that the analytical model under-estimated the actual lateral deflection. Table 4.6 lists the results of several selected shear walls. Table 4.5 Shear wall tests. 2x8x43x33-6-M1 4x8x43x33-6/12-M1 6x8x43x33-2/12-C1-C 2x8x43x33-6-M2 4x8x43x33-6/12-M2 6x8x43x33-2/12-C2-C 2x8x43x33-6-C1 4x8x43x33-6/12-C1 6x8x43x27-2/12-M1-D 2x8x43x33-6-C2 4x8x43x33-6/12-C2 6x8x43x27-2/12-C1-D 2x8x33x27-6/12-M1 4x8x33x27-6/12-M1 6x8x54x33-2/12-C1-B 2x8x33x27-6/12-M2 4x8x33x27-6/12-M2 6x8x54x33-2/12-C2-B 2x8x33x27-6/12-C1 4x8x33x27-6/12-C1 6x8x54x33-2/12-C2-C 2x8x33x27-6/12-C2 4x8x33x27-6/12-C2 2x8x43x33-4-M1 4x8x33x18-6/12-M2 2x8x43x33-4-M2 4x8x33x18-6/12-C1 2x8x43x33-4-C1 4x8x33x18-6/12-C2 2x8x43x33-4-C2 4x8x43x33-4/12-M1 2x8x43x33-2/12-C1-C 4x8x43x33-4/12-M2 2x8x43x33-2/12-C2-C 4x8x43x33-4/12-C1 2x8x33x27-2/12-M1 4x8x43x33-4/12-C2 2x8x33x27-2/12-M3 4x8x33x27-4/12-M1 2x8x33x27-2/12-C2 4x8x33x27-4/12-M2 2x8x33x27-2/12-C3 4x8x33x27-4/12-C1 4x8x33x27-4/12-C2 4x8x43x33-2/12-C1 4x8x43x33-2/12-C1-C 4x8x33x27-2/12-M1 Figure 4.7 Shear wall label definition for tests used. 27

37 Shear Wall Configuration Table 4.6 Comparison of lateral deflection. Aspect Ratio Deflection by Eq. 4.9 (in.) Deflection in Yu (21) and Yu and Chen (211) (in.) 2x8x33x x8x43x x8x33x x8x43x x8x43x x8x54x The selection of the three factors contributing to lateral deflection was based on the concept of the effective strip model. The assumptions included in the proposed deflection model are: the fasteners in the effective width of the sheathing experience the same tensile displacement and force, lateral deflection caused by the framing member deformation is ignored, deflection in the screws on the hold-downs is ignored, and the effect of the out-of-plane deformation to the lateral deflection is ignored. All those factors will contribute to the observed gaps from equation 4.9 and experimental results. Figure 4.7 shows a plot for the dimensionless lateral deflection difference ratio, difference between experimental results in Yu (21) and Yu and Chen (211) and results by equation 4.9, with the dimensionless ratio λ, defined in equation 3.6. A trend line was captured and this indicated that the proposed three factors respond positively to the design formula λ developed for the effective strip model. A fourth factor based on the effective strip model can now be introduced to improve the analytical model. 28

38 Δexp-Δlsw Δlsw λ Figure 4.8 Lateral deflection difference ratio verses λ for proposed model. 4.6 Complexity Factor CFS-SW-SS are known to be relatively complex structures to analyze, and developing equations which can accurately predict values for significant subjects such as strength and deflection can be a challenge. As mentioned earlier, the three factors that were initially considered to yield lateral deflection in CFS-SW-SS under-predicted the actual lateral deflection. A complexity factor, for all other effects that contribute to lateral deflection in CFS-SW-SS, was introduced to the lateral deflection model to obtain more accurate results. The factor was developed from the relationship between the dimensionless difference in lateral deflections ratio and the dimensionless slenderness ratio λ. Now, the analytical model including the complexity factor can be expressed as: llllll = 2 ff + Δ dddd cos αα + aaaaaah {1 + γγ(λλ) εε } (5.) where: γγ = scaling factor of complexity factor, 4.37; εε = exponent of complexity factor,

39 Table 4.7 compares the experimental peak lateral deflection results obtained from Yu (21) and Yu and Chen (211) with the peak lateral deflection results obtained by using equation 5.. As illustrated, the difference is with an acceptable range. Figure 4.8 illustrates the ratio of lateral deflections ratio verses λ. Most data points are within a reasonable range. The complexity factor was found to be effective since it covered the gap between the lateral deflection results predicted by the three factors initially considered and experimental lateral deflection results. Table 4.7 Comparison of lateral deflection results by the proposed model. Shear Wall Configuration Aspect Ratio Lateral Deflection in Yu (21) and Yu and Chen (211) (in.) Lateral Deflection by Eq. 5. (in.) 2x8x33x x8x43x x8x33x x8x43x x8x43x x8x54x Δexp Δlsw λ Figure 4.9 Lateral deflection ratio versus λ at peak load. 3

40 The model was further improved by introducing design equations to predict lateral deflection at shear strength benchmarks in 1% increments (to 1%). This was done to account for different shear strength degrees (V/Vn, V is the required shear strength determined by the designer and Vn is the nominal shear strength of the shear wall) desired by design engineers. Corresponding deflection values for fastener displacement, deformation in steel sheathing, and hold-down deformation at each shear strength degree were calculated and compared to lateral deflection results from Yu (21) and Yu and Chen (211). Figures show the relationship between the dimensionless lateral deflection difference ratio and λ for each degree. Practically, identical trends were observed at all degrees of shear strength. Δexp-Δlsw Δlsw λ Figure 4.1 Lateral deflection difference ratio versus λ, V/Vn =.1. 31

41 Δexp-Δlsw Δlsw λ Figure 4.11 Lateral deflection difference ratio versus λ, V/Vn = Δexp-Δlsw Δlsw Figure 4.12 Lateral deflection difference ratio versus λ, V/Vn =.3. λ 32

42 Δexp-Δlsw Δlsw Figure 4.13 Lateral deflection difference ratio versus λ, V/Vn=.4. λ Δexp-Δlsw Δlsw λ Figure 4.14 Lateral deflection difference ratio versus λ, V/Vn=.5. 33

43 Δexp-Δlsw Δlsw λ Figure 4.15 Lateral deflection difference ratio versus λ, V/Vn =.6. Δexp-Δlsw Δlsw Figure 4.16 Lateral deflection difference ratio versus λ, V/Vn =.7. λ 34

44 Δexp-Δlsw Δlsw Figure 4.17 Lateral deflection difference ratio versus λ, V/Vn =.8. λ Δexp-Δlsw Δlsw λ Figure 4.18 Lateral deflection difference ratio versus λ, V/Vn =.9. 35

45 Δexp-Δlsw Δlsw λ Figure 4.19 Lateral deflection difference ratio versus λ, V/Vn = 1.. Table 4.8 summarizes the terms γ, ε, σ for the complexity factor at the shear strength degrees considered. Fortunately, a neat linear and power relationship, illustrated in Figures 4.18 and 4.19, was captured between the shear strength degree and the scaling and exponent factors. Determining the scaling factor and the exponent factor for every shear strength degree were be simplified by using the equations generated from Figures 4.19 and 4.2. γγ = 2.51 VV (5.1) VV nn εε =.19 VV VV nn.266 (5.2) 36

46 Table 4.8 Summary list of scaling and exponent factors. Shear Strength Degrees, V/Vn Scaling Factor (γ) Exponent Factor (ε) Constant (σ) γ V/Vn Figure 4.2 Scaling factor verses shear strength degree. 37

47 ε V/Vn Figure 4.21 Exponent factor verses shear strength degree. The proposed analytical model for lateral deflection can now be expressed as shown equation 5.1. Note that since σ = 1., equation 5.1 can be simplified as shown. To use the proposed model, design engineers need to conduct connection strength tests to determine the fastener displacement values which correspond to their selected material and anchorage data specifications for their desired anchorage method. In addition, material properties need to be determined for the corresponding value of λ. llllll = 2 ff + Δ dddd cos αα + aaaaaah {1 + γγ(λλ) εε σσ} where: γγ = scaling factor of complexity factor; = 2 ff + Δ dddd cos αα + aaaaaah {γγ(λλ) εε } (5.1) = 2.51 VV VV nn ; εε = exponent of complexity factor; =.19 VV VV nn.266 ; 38

48 σσ = constant, 1.; VV = shear strength of shear wall selected by designer (plf); VV nn = nominal shear strength of shear wall (plf). Table 4.9 lists the lateral deflection results at peak load by equation 5.1 using the design equations introduced earlier. Appendix B provides two design calculation examples and lists the results for all shear wall specimens used in this research. The analytical model for lateral deflection yielded an acceptable estimate for lateral deflection. Figures illustrate the comparison for the lateral deflection at all selected shear strength degrees. A total of 47 shear wall test results were included in the analysis, and all ten strength degrees for each test were used to develop the complexity factor. Therefore, a total of 47 data points were considered in this research. Table 4.1 summarizes the statistical results for lateral deflection for all shear strength degrees. Overall, the proposed analytical model has good agreement with the test results, and it captures the non-linearity of the force vs. deflection behavior of the CFS-SW-SS. Table 4.9 Lateral deflection comparison by proposed model with design equations. Shear Wall Aspect Lateral Deflection in Yu (21) Lateral Deflection Configuration Ratio and Yu and Chen (211) (in.) by Eq. 5.1 (in.) 2x8x33x x8x43x x8x33x x8x43x x8x43x x8x54x

49 Table 4.1 Statistical analysis for proposed model. No. of Tests Shear Strength Degree, V/Vn Mean Std. Dev. COV All 47 Data Points exp lsw λ Figure 4.22 Lateral deflection difference versus λ using Eq. 5.1, V/Vn =.1. 4

50 exp 1. lsw λ Figure 4.23 Lateral deflection difference versus λ using Eq. 5.1, V/Vn = exp 1. lsw λ Figure 4.24 Lateral deflection difference versus λ using Eq. 5.1, V/Vn =.3. 41

51 exp 1. lsw λ Figure 4.25 Lateral deflection difference versus λ using Eq. 5.1, V/Vn =.4. exp lsw λ Figure 4.26 Lateral deflection difference versus λ using Eq. 5.1, V/Vn =.5. 42

52 exp lsw λ Figure 4.27 Lateral deflection difference versus λ using Eq. 5.1, V/Vn =.6. exp lsw λ Figure 4.28 Lateral deflection difference versus λ using Eq. 5.1, V/Vn =.7. 43

53 exp lsw λ Figure 4.29 Lateral deflection difference versus λ using Eq. 5.1, V/Vn =.8. exp lsw λ Figure 4.3 Lateral deflection difference versus λ using Eq. 5.1, V/Vn =.9. 44

54 exp lsw λ Figure 4.31 Lateral deflection difference versus λ using Eq. 5.1, V/Vn= Summary An analytical model for lateral deflection in cold-formed steel shear walls with steel sheathing under lateral loading is proposed in this research to predict lateral deflection. The proposed model uses design equations and provides a prediction for lateral deflection based on actual material and mechanical properties. It enables design engineers to predict lateral deflection at different desired shear loads. 45

55 CHAPTER 5 CONCLUSIONS AND RECCOMENDATIONS The primary objective of this research was to develop an analytical model based on the effective strip concept for the lateral deflection in cold-formed steel shear walls with steel sheathing (CFS-SW-SS) undergoing lateral loading. An existing model was reviewed to determine its applicability for shear walls with steel sheathing. An analytical model, equation 5.1, for lateral deflection was developed based on the four factors: fastener displacement, steel sheet deformation, and hold-down deformation, which are from the effective strip concept and a complexity factor, which accounts for the additional influential factors not considered in the previous three terms. 5.1 Connection Strength Tests In general, as the thickness of the specimens tested in the connection strength tests became larger, the peak load and displacement results were higher. Shear failure in fastener was only observed in specimen tests 1 and 2. Sheathing tear and build-up under the fastener was the dominant failure mode. The connection tests results were used to verify the previously established effective strip model for the nominal strength of CFS-SW-SS. 5.2 Proposed Analytical Model The proposed analytical model is essentially the extension of the effective strip model to address the deflection issue of CFS shear walls. The model is based on the real loading bearing mechanism of the shear wall and calibrated by the actual test results. Compared to the existing AISI deflection design model, the new model yields a better agreement with the test results and conforms to the actual behavior of the shear walls. The proposed method also allows the designers to be able to predict the shear wall s deflection at different loading levels. However, 46