bott masters thesis etd

8/13/2019 Bott Masters Thesis Etd

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HORIZONTAL STIFFNESS OF WOOD DIAPHRAGMS

by

James Wescott Bott

Thesis submitted to the faculty of

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

CIVIL ENGINEERING

APPROVED:

J. Daniel Dolan, Co-Chairman W. Samuel Easterling, Co-Chariman

Joseph R. Loferski

April 18, 2005

Blacksburg, Virginia

Keywords: wood diaphragm, shear stiffness, diaphragm stiffness, diaphragm deflection


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HORIZONTAL STIFFNESS OF WOOD DIAPHRAGMS

by

James Wescott Bott

ABSTRACT

An experimental investigation was conducted to study the stiffness of wood diaphragms.

Currently there is no method to calculate wood diaphragm stiffness that can reliably account for

all of the various framing configurations. Diaphragm stiffness is important in the design of wood

framed structures to calculate the predicted deflection and thereby determine if a diaphragm may

be classified as rigid or flexible. This classification controls the method by which load is

transferred from the diaphragm to the supporting structure below.

Multiple nondestructive experimental tests were performed on six full-scale wood

diaphragms of varying sizes, aspect ratios, and load-orientations. Each test of each specimen

involved a different combination of construction parameters. The construction parameters

investigated were blocking, foam adhesive, presence of designated chord members, corner and

center sheathing openings, and presence of walls on top of the diaphragm.

The experimental results are analyzed and compared in terms of equivalent viscous

damping, global stiffness, shear stiffness, and flexural stiffness in order to evaluate the

characteristics of each construction parameter and combinations thereof. Recommendations are

presented at the end of this study as to the next steps toward development of an empirical method

for calculating wood diaphragm stiffness.


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iv

TABLE OF CONTENTS

ABSTRACT ii

ACKNOWLEDGEMENTS iii

TABLE OF CONTENTS iv

TABLE OF FIGURES vi

TABLE OF TABLES viii

CHAPTER

I. INTRODUCTION 1

1.1 Introduction 11.2 Objectives and Scope of Research 41.3 Literature Review 7

1.3.1 Early Testing 71.3.2 Dynamic Testing 101.3.3 Similar Diaphragms 12

II. EXPERIMENTAL PROCEDURE 15

2.1 Scope of Testing 152.2 Test Apparatus 162.3 Diaphragm Construction 232.4 Test Parameters 272.5 Instrumentation 382.6 Test Protocol 452.7 Test Data Analysis 46

2.7.1 Yielding 472.7.2 Global Deformation 492.7.3 Cyclic Stiffness 512.7.4 Shear Deformation 56

2.7.5 Shear Stiffness 582.7.6 Flexural Deformation 602.7.7 Flexural Stiffness 602.7.8 Hysteretic Energy 612.7.9 Equivalent Viscous Damping 63

III. RESULTS AND DISCUSSION 67


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v

3.1 Introduction 673.2 Test Conditions 673.3 Nail Bending Test Results 693.4 Moisture Content and Density Results 713.5 Construction Parameter Results 73

3.6 Diaphragm Stiffening Methods 78

IV. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 82

4.1 Summary 824.2 Conclusions 83

REFERENCES 86

APPENDIX A 88

APPENDIX B 95

APPENDIX C 119

APPENDIX D 120

APPENDIX E 128


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vi

TABLE OF FIGURES

Figure Page

1.1 Cross-Section of a Typical Wood Framed Floor 2

1.2 Deep Beam Analogy 4

2.1 Load Frame, Actuator Connection, and Load Distribution Channel 17

2.2 Triangular Reaction Frame 18

2.3 Basic Test Apparatus and Configuration 20

2.4 Triangular Reaction Frame Plan View Schematic 21

2.4a Elevation View of Triangular Reaction Frame 21

2.5 Partial Section of Diaphragm Test Apparatus (loading parallel to joists) 22

2.6 Rim-Joist Splice for 10 x 40 ft. Specimens 24

2.7 Basic Specimen Sizes / Orientations (16 x 20 ft. and 20 x 16 ft.) 25

2.7 (Cont.) Basic Specimen Sizes / Orientations (10 x 40 ft.) 26

2.8 Fully Sheathed 10 x 40 ft. Specimen 27

2.9 Corner Sheathing Opening 29

2.10 Center Sheathing Opening 30

2.11 Test Configuration with Chords (and corner opening) 32

2.12 Test Configuration without Chords 32

2.13 Test Configuration with Walls 34

2.14 Wall-Lifting Davits 35

2.15 10 x 40 ft. Specimen with Walls and Wall-Braces 36

2.16 Application of Sprayed Foam Adhesive 37

2.17 Foam Adhesive Shown After Removal of a Sheathing Panel 38


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viii

TABLE OF TABLES

Table Page

3.1 Average Moisture Content and Density Results by Specimen 72

3.2 Average Percent Differences by Construction Parameter 74

A.1 Specimen 1 Test Results 89






B.1 Test Comparisons for the Effects of Blocking 96

B.2 Test Comparisons for the Effects of Foam Adhesive 99

B.3 Test Comparisons for the Effects of Blocking and Foam Adhesive 101

B.4 Test Comparisons for the Effects of Increased Nail Density 102

B.5 Test Comparisons for the Effects of Chords 103

B.6 Test Comparisons for the Effects of Walls 105

B.7 Test Comparisons for the Effects of Center Sheathing Openings 109

B.8 Test Comparisons for the Effects of Corner Sheathing Openings 113

C.1 Instrument Descriptions 119

D.1 Specimen 1 Joists Moisture Content and Density 121

D.2 Specimen 1 Sheathing Moisture Content and Density 122




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ix




E.1 Specimen 1 Test Descriptions 129







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CHAPTER I: INTRODUCTION 1

CHAPTER I

INTRODUCTION

1.1 INTRODUCTION

Modern structural engineering frequently involves sheathed construction, a load

resistance method exemplified in various structural elements by many combinations of suitable

materials. A common form of sheathed construction, the diaphragm, is a thin, usually planar

system of sheathing and frame members, intended to withstand considerable in-plane forces.

When referring to residential housing, everyday plywood construction typically comes to mind.

Most apparent examples of diaphragms are walls, upper-story floors, and roofs of

everyday structures such as residential houses, office buildings, and warehouses. Though similar

in function, wall diaphragms, called shear walls, require different consideration for design and

analysis, and thus fall outside of the scope of this investigation. Roofs and above-grade floors,

when designed as such, fall into the classification as true diaphragms. Typical combinations of

materials employed are wood sheathing on wood frame, metal sheathing on wood frame, metal

sheathing on metal frame, wood sheathing on metal frame, and variations using concrete,

structural insulation panels, and other construction materials. This research project is limited to

wood-framed and plywood-sheathed floor diaphragms typical in residential housing.

The common floor and roof diaphragm serves dual purposes by supporting vertical forces

(from loads such as furniture, people, snow, uplift, and its own dead load) and by transmitting

horizontal forces (from wind pressure or earthquake accelerations) to the supporting shear walls.


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Floors and roofs are inherently able to carry gravitational loads due to the typical design by

which appropriately spaced framing members are covered with sheathing and fastened together.

Joists and rafters, the common framing members of floors and roofs, respectively, are oriented to

maximize the moment of inertia for resistance to flexure. Thus, a 2x10 floor joist would be

installed such that the nominal ten-inch side is vertical. The sheathing spans the distance

between and transmits loads to the framing members below. The framing members then

distribute the loads proportionally to supporting walls or posts. Also, when adequately fastened

together, the sheathing and framing can produce a flexurally efficient composite section.

Resistance to vertical forces, though a primary consideration in design and construction of floors

and roofs, is not the subject of this study.

Sheathing

Wood Floor Joists

Figure 1.1 Cross-Section of a Typical Wood Framed Floor


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Horizontal forces applied to diaphragms are almost exclusively from wind and

earthquakes. Wind pressure on the exterior walls is transmitted proportionately along the edge

of a diaphragm as a uniform load. In the case of wind-loaded floors, connections with the top

plate of the wall below and the bottom plate of the wall above provide paths for load transfer.

When subjected to earthquake accelerations, its own inertia, or resistance to motion, and that of

attached walls or partitions causes horizontal loading of a diaphragm. Diaphragms are usually

more than capable of withstanding these loadings due to high in-plane shear capacity. Sheathing

material itself exhibits considerable in-plane shear strength. Hence, the reason a sheet of

plywood is much more rigid when loaded along the thin edge (in-plane) as opposed to the large

flat surface (out of plane). Accordingly, a low aspect ratio system of sheathing panels, properly

fastened together end-to-end along the edges, has an effective shear capacity. The fact that it is

usually so thin makes sheathing an efficient and lightweight means of resisting in-plane loads.

Resistance to these in-plane loads through diaphragm action may be compared to the

loading of a deep wide-flange beam, as illustrated in Figure 1.2. The shear walls of a structure

are analogous to the simple supports of the beam and provide the reaction against the forces

transmitted through the diaphragm. The inter-connected sheathing panels behave like the web of

the beam to resist the shear component of in-plane loads. And, the extreme edges of the

sheathing and/or the boundary members running perpendicular to the direction of the loads

simulate the flanges of the beam by carrying the tension and compression from the flexural

reaction to the loads.


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Flange

Web

Figure 1.2 Deep Beam Analogy

The behavior of floor and roof diaphragms has become an important issue with respect to

lateral stiffness and deflection. It has been noted that there is seldom a problem with the strength

of diaphragms, because failures are predominantly associated with the connections between a

diaphragm and supporting walls. Though the occurrence of actual failures is rare, diaphragms

have sometimes been a controlling factor in the overall failure of structures during seismic events

(Dolan 1999). Poor understanding of wood diaphragm behavior has spurred the interest of

researchers to formulate more accurate methods of analysis and design similar to methods

already employed in the design of cold-formed steel diaphragms.

1.2 OBJECTIVES AND SCOPE OF RESEARCH

The objective of this study is to evaluate the stiffness effects of various diaphragm

construction parameters for use in the development of an accurate method to determine shear


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stiffness of wood diaphragms based on the formulas currently used in cold-formed steel design.

Such a formula would allow an improved method of predicting diaphragm deflections. The

capability to accurately calculate diaphragm stiffness and deflections will enhance the safety and

economy of wood diaphragms.

The current lack of an accurate method to predict diaphragm stiffness prevents designers

from knowing exactly how much stiffness to expect from any given design. Adequate

diaphragm stiffness is required in order to allow load sharing among the supporting shear walls.

In other words, flexible diaphragms resist loads locally (i.e., they can not transfer loads

horizontally very far). Thus, the loads induced into a flexible diaphragm must be transferred to

local supports (i.e., walls that are the closest to the location of the induced load). A perfectly

rigid diaphragm would be the other end of the spectrum where all of the supporting walls share

in resisting the load according to their relative stiffness. In reality, the diaphragm stiffness falls

somewhere in between these two extremes. However, the higher the diaphragm stiffness, the

better the load sharing capability is of the structural system and therefore, the better the expected

performance.

Currently, diaphragms must be classified as either flexible or rigid in order to select one

of two different design methods for the transfer of load to the supporting structure. Wood

diaphragms have been traditionally assumed as flexible, thereby allowing a load distribution

design based on the tributary area method. This method, however, is not applicable when

diaphragms exhibit torsional irregularities such as asymmetric geometry and openings (e.g.

stairwells) or differences in locations of center of rigidity and center of force. To account for

such torsional irregularities a designer must assume a rigid diaphragm and transfer the load based

on relative stiffness of the supporting walls.


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According to the NEHRP Recommended Provisions for Seismic Regulations for new

Buildings and Other Structures (1997)and as adopted in the Uniform Building Code (1997)and

the 2000 International Building Codea diaphragms classification changes from rigid to flexible

when diaphragm deflection under load is equal to or greater than twice the deflection of the

supporting walls. Thus, designers are now forced to calculate the stiffness of wood diaphragms,

(and convert the stiffness into a corresponding deflection) in order to even make the distinction

between rigid and flexible.

Predicting diaphragm stiffness (or deflection) is difficult because currently there is no

simple and accurate method that can account for geometrical irregularities as well as all of

todays varying construction practices. The one current method for calculating deflection of

wood diaphragms as developed by APA is complicated and is not able to incorporate many

factors such as sheathing openings, absence of chords, use of sheathing adhesive, and non-

rectangular shapes. Designers need a simple and accurate method to determine wood diaphragm

stiffness if they are expected to even begin to select the proper load distribution method.

The specific objective of this study is to evaluate several basic diaphragm construction

details for their individual and combined effects on diaphragm stiffness. These results will then

be used under another task of the CUREE-Caltech Woodframe Project to develop and calibrate a

finite element model for diaphragm analysis. The overall goal of the experimental diaphragm

testing and finite element modeling is the development of an equation to accurately predict wood

diaphragm stiffness in the form of shear stiffness, G, as already accomplished by the cold-

formed steel industry.

This specific task is accomplished by a series of experimental tests on full-scale

diaphragms followed by careful analysis of the results. The test materials and procedures are


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discussed in detail in Chapter II. Data collected from the tests is evaluated in Chapter III for

numerical trends indicating the effects of the various specimen configurations on stiffness.

These trends are used to make general observations regarding the stiffening characteristics of

each diaphragm construction parameter. All of the stiffness results for every test of each

specimen are listed for reference in Appendix A.

1.3 LITERATURE REVIEW

The objective of this literature review is to examine the studies in the field of theoretical

and experimental diaphragm research. A study of research performed in this field is important in

order to know what characteristics have already been established and what questions of

diaphragm behavior are still unanswered. Sporadic since the 1950s, most of the testing of wood

diaphragms has occurred at the facilities of the Douglas-fir Plywood Association (DFPA),

American Plywood Association (APA), Oregon State University, Oregon Forest Products

Laboratory, Washington State University, and West Virginia University. The volume of

literature available is small, therefore, rather than place the review in a separate chapter, the

reviewed literature is provided as a section within this chapter.

1.3.1 Early Testing

The DFPA sponsored some early tests in diaphragm behavior. Countryman (1952)

describes lateral tests on plywood-sheathed diaphragms. Four specimens, 12 x 40 ft. and 20 x 40

ft., and six one-quarter scale models, 5 x 10 ft., were tested by monotonic loading at fifth-points.

The specimens had varying parameters such as blocking, openings, staggered panels, gluing,

plywood thickness, nail size, and boundary nailing patterns. Stiffness of the diaphragms was

calculated from measured lateral deflection in the middle of the lower chord and applied load.


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Shear deformation, and not flexural deformation was determined to be the predominant form of

deflection. Load versus deflection plots show that the actual deflection was consistently higher

than calculated values using existing equations. It was found that diaphragms behave like a

horizontal girder with a shear-resistant web. Due to their location at the extreme edges, chord

members of a diaphragm act like the flanges of a girder by resisting the flexural tension and

compression forces. The sheathing serves as the web of the girder to resist the shear. Strength

and stiffness of the specimens was found to be primarily dependent on the strength of the nailed

plywood-to-frame connections.

Due to over conservative design codes, the DFPA pursued further studies in diaphragm

action. Countryman and Colbenson (1954) report on tests of fifteen full-scale diaphragms,

conducted to better understand the strength effects from:

1. Omission of blocking

2. Panel arrangement

3. Nailing schedules

4. Span-thickness combinations

5. Length-width ratio

6. Seasoning of frame lumber

7. Use of three inch lumber

8. Cut-in blocking for chords

9. Load application perpendicular to joists

10.Screwed cleats in lieu of blocking

All 24 x 24 ft. specimens were monotonically loaded with four equal lateral forces at fifth

points of the span, and deflections were measured from the middle of the unloaded chord.

Plywood thickness and nailing schedule, along with blocking to a lesser degree, were found to be

the predominant factors in determining strength and stiffness. Ultimate applied shears ranged


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from 733 to 2530 plf, while ultimate deflections occurred from 0.52 to 3.2 in. For blocked

diaphragms, the measured deflections are consistent with a formula produced as a result of the

DFPA Report No. 55 (Countryman 1952) with an average error of 15%.

In conjunction with the research described above, the DFPA also sponsored tests at the

Oregon Forest Products Laboratory. The two 20 x 60 ft. roof diaphragms tested, were

constructed with the lightest framing and plywood thickness permissible at that time for a roof of

this size (Stillinger and Countryman 1953). The 2 x 10 joists were framed at 24 in. o.c. and

sheathed with 3/8-in. thick plywood. One of diaphragms was blocked along the panel edges.

The diaphragms were loaded monotonically by hydraulic jacks at the fifth points. The 3/8-in.

thick plywood was found to be adequate, though not as strong as specimens with thicker

plywood. The lightweight framing system performed adequately. Lastly, it was found that for

unblocked diaphragms, no special boundary nailing detail was required regardless of the reduced

strength.

The APA became interested in lateral shear testing of diaphragms not composed of

Douglas-fir plywood. Tissell (1966) validated the DFPA tests from 1955 as well as going on to

test diaphragms of other various species of wood that were becoming popular in construction.

Nineteen full-scale 16 x 48 ft. diaphragms were tested. Plywood characteristics, sheathing-to-

framing connections, nail types, and framing member types were varied in the tests. Monotonic

loading from 16 hydraulic jacks at 3 ft. on center was used to approximate a uniform lateral load.

Lateral deflections were measured with dial gages at mid span of the tension chord. The design

shear values were found to be very conservative, with the average ultimate load being 1545 plf

and the average allowable design load being only 420 plf (includes factors of safety). Sheathing

of different species of wood was found to have a small but accountable change in shear strength


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and stiffness. However, effects from plywood grade and quality were found to be negligible.

Tissell concluded that shear strength equivalent to that of blocked diaphragms is possible by

stapling tongue-and-groove 2-4-1 plywood. Further, shorter ring-shank nails are permissible as

long as a minimum penetration is attained. Open-web steel joist-framed diaphragms were

slightly stronger than the lumber framed diaphragms. The DFPA design values determined from

the tests previously discussed (Countryman and Colbenson 1954) were found to adequately

conservative.

1.3.2 Dynamic Testing

GangaRao and Luttrell (1980) explain the efforts at West Virginia University to quantify

shear response of diaphragms with the ultimate goal being the preparation of accurate analysis

models for future design purposes. Since diaphragms had been mainly studied under static

loading conditions, they propose that stiffness characteristics are an equally critical issue in a

correct estimation of behavior under real-life dynamic loading. Preliminary dynamic results

from tests at West Virginia University were used to derive joint slip and shear deformation

response equations based on dynamic loading. They predicted that damping characteristics with

respect to joint slip are the critical factors needed to appropriately describe diaphragm behavior

under dynamic loads.

At the time, Polensek (1979) was the only researcher making attempts at quantifying

damping characteristics for horizontal dynamic loading. His tests of plywood sheathed

diaphragms with six or ten inch joists yielded average equivalent viscous damping ratios

between 0.07 and 0.11. However, he considered that the data accumulated had been too varied

for an accurate estimation of the damping ratio. It was apparent, however, that an increase in the

damping effect is directly proportional to floor span.


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At West Virginia University, Jewell (1981) performed experimental tests on partial

(cantilever) diaphragms in order to analyze a range of different parameters such as nail spacing,

boundary conditions, connection details, load type, and damping capacity. Three 16 x 24 ft.

diaphragms and six 16 x 16 ft. diaphragms were tested under monotonic, cyclic, and impact

loads in the directions perpendicular and parallel to the joists. Replica diaphragms were also

modeled in the same configurations as flexible composite members in a finite element analysis to

determine any inaccuracy in this theoretical approach. Based on a comparison of the theoretical

and experimental test results, Jewell was able to analyze relationships of plywood behavior, nail

slip, effect of loading, effect of joist hangers, and damping to the stiffness of diaphragms. In

most cases, the finite element approach yielded slightly less conservative results for stiffness

(i.e., predicted deflections were lower than actual), based on the parameters listed above.

Corda (1982) and Roberts (1983) performed additional cantilever diaphragms tests at

West Virginia University in another codependent study involving laboratory testing and finite

element modeling. Corda tested six 16 x 24 ft. specimens cyclically and statically to failure in

order to study local and global in-plane shear stiffness response to variations of blocking,

openings, plywood thickness, corner stiffeners, and framing nail sizes. It is noteworthy that nail

softening after loads up to 9 kips on some specimens caused a large decrease in stiffness.

Increased plywood thickness (without using longer nails) and corner openings reduced strength

but had little effect on stiffness. Roberts theoretical analysis of equivalent models of the

diaphragms tested by Corda, showed some evident discrepancies. Problems with the finite

element analysis program included the limitation to monotonic loading, inaccurate predictions of

panel slip, and the iterative processes of calculation of diaphragm deflection with respect to nail

slip, a bilinear relationship, demanding the modification of results to a nonlinear solution using


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the tangent stiffness method. Based on the problems encountered, Roberts suggested that the

limitations imposed on the program user in modeling plywood diaphragms need to be eliminated

by further experimental research into stiffness characteristics of panel slip, plywood layout and

connection, diaphragm openings, and nail slip under cyclic loading.

A recent APA report by Tissell and Elliott (1997) describes diaphragm testing for high

load conditions equivalent to earthquakes accelerations. The primary intent was to formulate

design and construction approaches for these high-load diaphragms, which may incorporate

use of two layers of plywood, thicker plywood, or stronger fastener conditions. Ten of the

diaphragms tested were 16 x 48 ft., and the dimensions of an eleventh specimen were changed to

10 x 50 ft. Hydraulic jacks at a spacing of 24 in. o.c. were used to apply a cyclic uniform load

along the long side of the diaphragms. Results show that it is possible to increase shear strength

by increasing the number of fasteners or adding another layer of sheathing in areas of high shear.

This report also notes that plywood panel shear capacity must be checked for high-load

diaphragms. Staples were found to be adequate fasteners in lieu of nailed sheathing-to-framing

connections. Along the same lines, field glued joints and a reduced number of nails are adequate

for these diaphragms.

1.3.3 Similar Diaphragms

The abundant studies of floors comprised of materials other than wood are important in

order to understand general behavior of diaphragms. It is possible that wood diaphragm design

methods may be simplified and accurately rationalized in terms of methods already in use for

other materials. A theoretical study of the behavior of composite steel beam and concrete deck

diaphragms was made by Widjaja (1993) at Virginia Polytechnic Institute and State University.

Similar to many efforts currently in progress for wood diaphragms, the purpose of this study was


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to develop an accurate finite element analysis model that predicts diaphragm behavior,

incorporates possible variations of design parameters, and derives design strength equations.

Similarly, experimental cantilever tests on cold-formed steel diaphragms are important in the

design of many steel-frame building roofs and composite floors (during construction, before

concrete). Post-frame diaphragm testing (wood frame and metal sheathing) is also significant in

terms of the more agricultural or shed type buildings.

With sponsorship from NUCOR Research and Development, Hankins et al. (1992)

performed eighteen cantilever diaphragm tests at Virginia Polytechnic Institute and State

University to determine strength and stiffness of cold-formed steel sheathing, 20 and 22 gage

thickness (Vulcraft 1.5B1 deck), welded or bolted to a steel frame. The 16 x 16 ft. specimens

were subjected to monotonic loads. Thirteen diaphragms utilized an 8 ft. span, requiring only

one filler beam. The other five specimens had a filler beam spacing of 4 ft., requiring three filler

beams. Bolt and puddle weld arrangement, used to secure the sheathing, was varied to determine

its effects on diaphragm behavior. Results from the tests indicate that specimens with thicker

gage sheathing have more strength and stiffness. However, even though specimens with smaller

filler beam spacing (three filler beams as opposed to one) had more strength, the diaphragm

stiffness was less in some cases.

Hausmann and Esmay (1977) report the results of tests on twenty-six full size post-frame,

metal-clad diaphragm panels. All specimens were 8 x 16 ft. with rafters at 4 ft. o.c. along the 16

ft. side and purlins at 2 ft. o.c. along the 8 ft. side. Loaded monotonically at the ends of the three

interior rafters, the panels were analyzed for strength and stiffness based on varying parameters

such as framing arrangement, type, number, and metal of fasteners, aluminum or steel sheathing,

and with or without insulation. It was determined that purlins laid flat to the rafters was the more


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suitable method of framing. Screw fasteners in the panel valleys increased diaphragm stiffness

and strength, especially for the steel clad specimens. Aluminum panels were more suitable with

nailing, due to a larger cover width for each sheet. Placing insulation between wood-framing

and metal cladding not only reduces diaphragm strength, but also seriously affects stiffness.

Fastener configurations have important and measurable effects on diaphragm behavior.

Anderson and Bundy (1990) performed additional post-frame diaphragm tests to outline

the effects of openings in the sheathing. Fifteen cantilever specimens, 7-2/3 x 12 ft. with two

interior rafters and seven purlins were tested monotonically with varying amounts of sheathing

missing. Diaphragms were constructed with SPF lumber, screw fasteners, and steel sheathing.

Fastener configurations were found to be extremely important for diaphragm stiffness.

Openings in the sheathing at normal intervals caused the specimens to be ineffective as

diaphragms. It was also found that spacing of purlins has little impact on strength or stiffness of

the diaphragms.

In addition to the physical testing, there has been a great deal of computer modeling of post-

frame diaphragms for scientific purposes in order to aid designers and validate experimental

results. For example, Wright and Manbeck (1993), among many others, conducted finite

element analyses of post-framed diaphragm panels. Following procedures provided by Woeste

and Townsend (1991), they modeled full size 8 x 12 ft. diaphragms with 2x4 in. purlins at 2 ft.

o.c., 2x6 in. rafters at 3 ft. o.c., and steel cladding secured with 16d nails. The finite element

model was compared to three identical experimental diaphragm tests. The finite element model

closely predicted diaphragm shear strength, but under-estimated shear stiffness by 28%. Results

show that discrepancies arise due to difficulties in modeling nonlinear behavior of fasteners and

intricate load paths between the wood frame and steel sheathing.


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CHAPTER II: EXPERIMENTAL PROCEDURE 15

CHAPTER II

EXPERIMENTAL PROCEDURE

2.1 SCOPE OF TESTING

Multiple stiffness tests and one test to failure were performed on each of six full-scale

diaphragms at the Thomas M. Brooks Forest Products Center of the Virginia Polytechnic

Institute and State University located in Blacksburg, Virginia. Consortium of Universities for

Research in Earthquake Engineering (CUREE) sponsored the research under its Wood Frame

Project, Task 1.4.2 Diaphragm Studies.

Diaphragm dimensions for four specimens were 20 x 16 ft. with varying orientations,

while two high-aspect ratio specimens were 10 x 40 ft. Multiple tests on each specimen were

possible due to the non-damaging deflections being imposed, allowing an economical means of

incorporating multiple test parameters. Test parameters investigated for effects on diaphragms

stiffness were: 1) corner opening, 2) center opening, 3) fully sheathed, 4) 6-12 nail pattern, 5) 3-

12 nail pattern, 6) with/without chords, 7) with/without walls, 8) with/without blocking, and 9)

with/without foam adhesive. Specimens were subjected to non-destructive, low-amplitude

dynamic-cyclic loading by a computer-controlled hydraulic actuator, while load and deflection

values were being recorded by a computer data acquisition system. The final test on each

specimen, though not a primary focus of this study, was an attempt to cause diaphragm failure.


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2.2 TEST APPARATUS

Diaphragm testing was conducted on a 22 X 50 ft. concrete pad with 42 in. wide by 27 in.

tall concrete back-walls along two adjacent sides. The heavily reinforced back-walls have two

7/8 in. diameter anchor bolts embedded in the concrete at 2 ft. on center with a 400,000 lb point-

load capacity at a minimum spacing of 6 feet.

A computer controlled hydraulic actuator was mounted horizontally at the midpoint of

the 50 ft. back-wall. The actuator had a 55 kip capacity with a 6 in. stroke, and included a 50

kip Interface load cell, screwed onto the end of the hydraulic cylinder. Load was transferred

from a ball joint at the end of the actuator, through a pin-connected gusset to a 20 ft. long

C6x10.5 steel channel. The channel, as shown in Figure 2.1, was fastened along the entire width

in the center of the specimen span with 5/8 in. diameter lag screws. In cases where a joist did not

fall in the center of the diaphragm, 4x4 blocks are placed under the sheathing to provide a

backing for the lag screws used to attach the steel load distribution channel.


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Figure 2.1 Load Frame, Actuator Connection, and Load Distribution Channel

Offset equal distances (based on dimensions of diaphragm specimens) from the centerline

of the actuator were triangular reaction frames, as shown in the photograph of Figure 2.2. The

frames were constructed of 4 X 6 in. steel tubes welded together. Each reaction frame was


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connected to the concrete back-wall with four 7/8 in. diameter anchor bolts. A one-inch thick

end plate was welded to the end of the steel tube of the reaction frame and was drilled and tapped

for a 1 in. threaded rod. A two-foot piece of threaded rod was screwed through the plate into

the steel tube leaving the desired length exposed. A shop-fabricated, full-bridge load cell made

of 2 in. diameter steel rod and strain gauges screwed onto the opposite end of the threaded rod.

The load cell had a large hex-nut welded to one end to connect to the threaded rod protruding

from the triangular reaction frame. Gusset plates welded to the opposite end of the load cell

provided a pinned connection to the diaphragm support frame.

Figure 2.2 Triangular Reaction Frame

Each end of the diaphragm was attached to a 20 ft.-L2x2x steel angle, which was

welded intermittently to the side of a 20 ft.-3 X 5 in. steel tube using lag screws. One end of

each steel tube was pin-connected to the gusset plates of the reaction load cells. This support


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frame served the same purpose as shear walls by transmitting loads out of the diaphragm at each

end. The reaction frame load cells then measured these loads. The support frame served a

secondary purpose, to hold the diaphragm at the proper elevation for concentric loading from the

actuator. Several one-inch diameter PVC pipes were also placed on the concrete under the

specimen to help hold the interior of the specimen at the proper elevation, and to act as

frictionless rollers under the joists as load was applied. The schematic drawings of Figures

2.3, 2.4, and 2.5 illustrate all of the elements of the test apparatus including the specimen support

frame, the triangular reaction frame, and the load distribution channel.


29/144


Figure 2.3 Basic Test Apparatus and Configuration

Steel Channel(C6 x 10.5)

Lag Screws

Concrete Back Wall

See Figure 2.4

for Detail

Fig.2.5

Load Cell

Actuator

Diaphragm Sizeand SheathingLayout Varies

Support Frame3" x 5" Steel Tube

Triangular ReactionFrame

Load Cell


30/144


Load Cell

Steel TubeFrame

Fig.2.4a

Figure 2.4 Triangular Reaction Frame Plan View Schematic

Pinned Connection

ConcreteBackwall

Diaphragm Support Frame

Figure 2.4a Elevation View of Triangular Reaction Frame


31/144


Figure 2.5 Partial Section of Diaphragm Test Apparatus(for specimens loaded parallel to the joists)

Steel Angle:-Welded to Steel Tube-Lag Screwed to End Joist

2 x 12 Joists @ 16"

(Douglas Fir)

2332" T&G Plywood

Sheathing

PVC Pipe Rollers

114"

Steel Channel (C6X10.5) - LagScrewed to non-structural blocks

Steel PipeRoller

3" x 5" x 3 8" Steel Tube


32/144


2.3 DIAPHRAGM CONSTRUCTION

Of the six, full-scale diaphragm specimens, two were 16 x 20 ft. in dimension and were

loaded parallel to the direction of the joists on the 20 ft. side. Two specimens were 20 x 16 ft. in

dimension, loaded perpendicular to the joists on the 16 ft. side. The last two specimens were 10

x 40 ft. in dimension, loaded parallel to the joists on the 40 ft. side. Resembling the size and

shape of one side of a roof of a typical residential home, these 10 x 40 ft. specimens were

intended to test the envelope of diaphragm performance with respect to aspect ratio.

The diaphragm specimens were framed with Douglas-fir 2 x 12 joists spaced at 16 in. o.c.

and nailed with three 16d nails at each end to a 2 x 12 Douglas-fir rim joist. In the case of

specimens loaded parallel to the direction of the joists, the bottom of each end joist was attached

to the diaphragm support frame using lag screws as shown in Figure 2.5. Conversely, when

loading was applied perpendicular to the joists, the rim-joists were connected to the support

frames. Since the lumber used was 20 ft. in length, the rim joists of the 40 ft. long specimens

had to be spliced in the center with steel plates and bolts as shown in Figure 2.6 (while such a

splice may not be feasible in real-life construction due to interference with finish materials,

effective transfer of compression and tension forces in the chords was essential for valid test

results). The three main specimen configurations, including loading and reaction locations, are

schematically illustrated in Figures 2.7a-c. A photograph of a 10 x 40 ft diaphragm specimen is

presented in Figure 2.8.

A wood sample was taken from each joist of every specimen for moisture content and

density analysis. This information was recorded for possible use when evaluating test results,

since moisture content changes in lumber affects fastener performance.


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Figure 2.6 Rim-Joist Splice for 10 x 40 ft. Specimens (typical both sides)


34/144


(a) 16 x 20 ft. Specimen

(b) 20 x 16 ft. Specimen

Figure 2.7 Basic Specimen Sizes / Orientations

20'

Load Applied4'x8'x23 32" T&G

Plywood Sheathing

16'

Cut-out showsframing layout below

2 x 12 Joists @ 16"(Douglas Fir)

2 x 4 Blocking(on flat)

2 x 12 Rim-Joist(Douglas Fir)

20'

Load Applied

16'


35/144

Load Applied4'x8'x2332" T&G P

Sheathing

40'

(c) 10 x 40 ft. Specimen

Figure 2.7 (Continued) Basic Specimen Sizes / Orientations


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Figure 2.8 Fully Sheathed 10 x 40 ft. Specimen

The specimens were sheathed with nominal 4 x 8 ft. sheets of 23/32 in. tongue-and-

groove plywood in a staggered panel configuration. Sheathing was cut as required to complete

the desired panel configuration. Sheets were attached to the framing with 10d nails in a 6/12 nail

pattern, meaning nails are spaced at 6 in. around the perimeter and at 12 in. on the interior

supports of each sheathing panel. Typical sub-flooring construction adhesive was not used

between the joists and plywood sheathing; however some tests involved the use of sprayed foam

adhesive.

2.4 TEST PARAMETERS

Specimens were subjected to a number of different construction variations, including the

multiple combinations thereof. The variations tested were:

1. Sheathing openings fully sheathed, corner opening, center opening


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2. Chord members with / without rim joist

3. Blocking with / without 2 x 4 blocking

4. Walls with / without 4 ft. tall stud-framed walls

5. Sprayed foam adhesive & nails versus nailed only

6. Sheathing nail density 6/12 versus 3/12 nail pattern

Variations to the basic specimen listed above, are individually discussed in detail in the

following paragraphs.

Openings in the plywood sheathing were intended to simulate common openings in floors

of residential homes for stairways, atriums, and vaulted ceilings. These openings weaken and/or

cause torsional irregularities that can dramatically affect the stiffness of diaphragms. Duplex

(double-headed) 10d nails were used to fasten the sheathing panels that were to be removed from

the specimens in order to simulate openings. The corner opening in all sizes and orientations of

specimens was easily achieved by removing one full 4 x 8 ft sheet of plywood from a corner.

However, the center opening presented more challenges due to the staggered sheathing

configuration and tongue-and-groove plywood. For both orientations of the 16 x 20 ft.

specimens, an 8 x 12 ft. rectangular opening was made by prying the unfastened sheets up in the

center along the tongue-and-groove seam like an army tent and lifting them out. Some sheets

had to be cut in half to achieve a rectangular opening. Due to the high aspect ratio of the 10 x 40

ft. specimens, a proportional rectangular opening in the center was not reasonable, since typical

roof and floor diaphragms would not have such an opening. A schematic drawing and

accompanying photograph of the two opening types used in the tests are presented in Figures 2.9

and 2.10.


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Figure 2.9 Corner Sheathing Opening

Load Applied


39/144


Figure 2.10 Center Sheathing Opening

Load Applied


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The chords of a diaphragm are the exterior framing members that are oriented

perpendicular to the direction of loading. They serve to resist bending moment induced in a

diaphragm while also supporting the extreme edges of the sheathing. In the case of floor

diaphragms, the chords may either be the rim joist or simply the last joist at each end of the floor,

depending on the orientation and direction of loading. Residential roof diaphragms typically do

not have a true rim joist, either at the lower edge along the fascia or at the ridge (unless the fascia

board or ridge beam is considered to be effective). The absence of an effective chord is

especially prevalent for roof systems utilizing metal plate connected trusses.

Though all testing was performed on floor diaphragm specimens, chord effects should be

similar for roof-like specimens. The effectiveness of chords was quantified by running tests with

and without the designated chord members (rim joists) in place. Only those specimens having

rim joists as the chords (specimens loaded parallel to the direction of the joists) could be tested in

this manner. The rim joists were nailed to the diaphragm at each joist with three 16d duplex

nails. Plywood edges were nailed to the rim joist with 10d duplex nails at 6 in. o.c. Duplex nails

were used for easy removal of the rim joists between different test specimen configurations. One

of the diaphragm specimens is shown in both configurations of having the rim joist acting as the

chord in place and removed in Figures 2.11 and 2.12 respectively.


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Figure 2.11 Test Configuration with Chords (and corner opening)

Figure 2.12 Test Configuration without Chords


42/144


Blocking is the term used for the short framing members that span between joists and

serve to interlock the unsupported joints between sheathing panels. They can have the same

cross-sectional dimensions as the joists (full-depth blocking improves noise and vibration

dampening), or blocks can simply be smaller lumber laid on flat. In this investigation, specimen

configurations with blocking used 2 x 4s laid on flat installed between joists where each line of

unsupported sheathing panel joints would fall prior to installing the sheathing. The blocks were

fastened to the joists on each side with two 16d common toe nails. Plywood panel edges that fell

over the blocks were nailed every 6 in. with 10d duplex nails for easy removal to simulate

blocked and unblocked conditions. To reconfigure the specimen without blocking, the sheathing

nails were extracted, the diaphragm was tilted on-end with a forklift, and blocks were removed.

Likewise, replacing blocking involved tilting the diaphragm up to install new 2 x 4 blocks from

below, and then re-nailing the sheathing to the blocking.

A potentially significant unknown in diaphragm design is the effect of walls on the

horizontal stiffness. Walls of a structure transfer wind loads to the floors to which they are

connected. Also, the mass of the walls themselves present added lateral loads to diaphragms

during earthquakes. These walls, especially the flexural stiffness of their own bottom plate, may

also benefit a floor diaphragm by helping to resist these same lateral loads.

For testing purposes, four-foot high walls were installed along the two chord edges of

specimens. As shown in Figure 2.13 these walls were constructed of 2 x 4 studs at 16 in. o.c.

and 7/16 in. OSB sheathing on the outside. The 2 x 4 bottom plate of the walls was fastened

through the plywood sheathing to the floor joists below with 3 x in. self-tapping Simpson

screws for a strong connection yet easy removal. For the 16 x 20 ft. specimens, regardless of

orientation, the walls were set in place and removed with a long, cable-supported, boom attached


43/144


to a large forklift. Starting with the second specimen, lever-action davits, as shown by the

photographs of Figure 2.14, were welded to the side support frames at each end of the walls to

more quickly and safely facilitate raising and lowering for configurations with and without walls.

As shown in Figure 2.15 for the 40 ft. long specimens, the walls on each side were built in 20 ft.

sections, set in place with the boom, and connected in the center. From then on, the davits at

each end of the walls accompanied by braces in the center allowed for repeated installation and

removal of walls. Braces were required to laterally stabilize the wall segments for the 10 x 40 ft.

diaphragm specimens.

Figure 2.13 Test Configuration with Walls


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(a) Walls Lowered and Attached (b) Walls Unfastened and Raised

Figure 2.14 Wall-Lifting Davits


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Figure 2.15 10 x 40 ft. Specimen with Walls and Wall-Braces

Current trends in construction involve the use of adhesives for an ever-widening range of

applications. In this case the method of fastening sheathing panels to framing members was

varied between nailed only and nailed plus sprayed adhesive. The adhesive material used was a

sprayed, two-part, self-expanding, poly-isocyanurate foam adhesive manufactured by ITW

Foamseal. The foam adhesive was tested using coupon tests to quantify its stiffness as a

connection. The connection stiffness was determined to be equivalent to that obtained by using

elastomeric adhesives typically used in wood floor construction. While elastomeric adhesive

would have been more representative of traditional construction, it would have prevented the

possibility of removing sheathing without damage once fastened down, thereby making it costly

and difficult to alter specimen sheathing configurations. Sheathing fastened down with the foam

adhesive could be removed with minimal damage by cutting the adhesive at the joints with a


46/144


knife. Also, testing of the foam adhesive will provide useful information for its effectiveness in

roof retrofit applications. After having tested all of the nailed-only configurations, the foam

adhesive was applied to the underside of fully-constructed specimens that could be safely tilted

on-end (i.e. the two 16 x 20 ft. and the two 20 x 16 ft. specimens). Specifically, the adhesive was

sprayed along each side of every joist at the interface with the sheathing. Adhesives were not

used on the 10 x 40 ft. specimens due to the specimens flexibility, which made tilting the

specimens without damage impossible. A photograph of the foam adhesive being applied is

shown in Figure 2.16, and a photograph of a sheathing panel removed is shown in figure 2.17.

Figure 2.16 Application of Sprayed Foam Adhesive


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Figure 2.17 Foam Adhesive Shown After Removal of a Sheathing Panel

Sheathing nail density was varied for a few tests on the third and fourth specimens, both

of which were 20 x 16 ft. loaded perpendicular to the joists. The nail pattern was changed from

6/12 to 3/12 on the fully sheathed and nailed only configurations. In other words, nail spacing

around the perimeter of each sheathing panel (where supported by joists or blocking) was

decreased from 6 in. to 3 in. o.c. using easily removable 10d duplex nails. The 3-12 nail pattern

was tested while the walls and blocking parameters remain variable.

2.5 INSTRUMENTATION

Movements, deflections, and loads were measured at multiple locations on the diaphragm

specimens using electronic sensors of various types in conjunction with a computer controlled

Data Acquisition system (DAQ). The DAQ used for this project LABTECH, a Windows PC-


48/144


based program. Prior to testing, all instruments used in this research project were carefully

calibrated for accurate results. Before any series of tests in a day, all instruments were checked

to make sure they were correctly mounted, functioning properly, and zeroed. Due to the exposed

conditions of the outside testing facility, all instruments were demounted and taken inside or

covered with plastic daily to protect from inclement weather, dew, and frost.

An internal Linear Variable Displacement Transducer (LVDT) measured the deflections

caused by the hydraulic actuator. Signals from this highly sensitive device were transmitted to

the computer controller, which in turn used the information as the feedback channel to control

the actuator. Loads measured by the 50 kip Interface load cell were recorded by the DAQ and

had no effect on the displacement-controlled actuator.

The custom-built load cells at each end of the diaphragm measured the reaction loads,

both in tension and compression due to the cyclic loads from the actuator. These reactions

simulated the shear loads that supporting walls of an actual structure must withstand. Prior to

use, these load cells, as described in Section 2.2, were separately calibrated in tension only on a

universal testing machine with an excitation of 10 Volts (compression was not feasible due to a

pinned-pinned condition when using special calibration fixtures). Both were loaded

incrementally to 40 kips tension, and in each case the linear calibration plot proved that no

yielding within the load cell occurred. The slopes of these lines were used in the DAQ as

multipliers to convert the output voltage signals from the load cells into equivalent values of

load.

Horizontal movement of the plywood sheathing relative to the framing members below

was measured at two locations with external LVDTs. An aluminum bracket mounted to the

end-joist at each rear corner held the barrels of a pair of LVDTs in place horizontally. The


49/144


plungers of these instruments react against metal tabs, which were screwed and glued to the

plywood at the rear corners of the diaphragm. The LVDTs of each pair pointed in orthogonal

directions to account for biaxial sheathing movement. A photogragh of the LVDT mounting

setup is shown in Figure 2.18.

Figure 2.18 LVDTs and Mounting Bracket


50/144


String Potentiometers (abbreviated, string-pot) were used at multiple locations to

determine diaphragm movement, deformation, and slippage relative to the test frame. Seven

string-pots were set along the front face of the specimens to measure the global deflection. A

string-pot was attached to each steel side support frame to determine the slip in the side load cell

connections and between the steel frame and the diaphragm itself. Likewise, a string-pot was

mounted to the steel load channel in the center to determine any slip its lag screw connection to

the specimen. Two string-pots were mounted diagonally on each side of the diaphragm

centerline to record the deformation caused by shear deflection during testing. Schematics

illustrating the positions of each displacement sensor for each specimen configuration are

presented in Figures 2.19 through 2.21. A list describing each instrument is presented in

Appendix C.


51/144


GR1

Concrete Back Wall

LVDTL-NS

LVDTL-EW

GL1SlipL

DL1

SlipC

DL2 DR1

GL2 GL3 GC

LVDT

R-NS

LVDTR-EW

DR2

3" x 5"Steel Tube

GR2 SlipRGR3

Figure 2.19 Instrumentation Plan for 16 x 20 ft. Specimen


52/144


Concrete Back Wall

GC GR1 GR2 GR3 SlipRGL1 GL2 GL3SlipL

SlipC

DL1 DL2 DR1 DR2

LVDTL-NS

LVDTL-EW

LVDTR-EW

LVDTR-NS



53/144

Concrete Back Wall

GC GR1 GR2GL1 GL2 GL3SlipL

SlipC

DL1 DL2 DR1

LVDT

L-NS

LVDT

L-EW



54/144


2.6 TEST PROTOCOL

In most cases of experimental research, how a specimen is stressed and to what extent, is

equally as important as the specimens characteristics. In this project great care was taken to

NOT apply deflections so large that specimens reached or exceeded their yield point and were

damaged. On the other hand, it is critical to apply sufficient deflection to obtain valid test data.

This limit was found for each specimen size and orientation by loading monotonically in small

increasing increments until signs of diaphragm damage are seen or heard, or until the slope of the

load/deflection curve, shown in real-time on the DAQ computer screen, appeared to be

decreasing. The deflection amount used for all tests was slightly lower than the largest

monotonic deflection. Additionally, deflections for this project followed a cyclic pattern that

somewhat simulates the cyclic loading of earthquakes, only not nearly as rapid, since this

apparatus is not intended or equipped to perform shake-table testing.

The load protocol for all tests, except those to failure, was five sinusoidal cycles at the

predetermined deflection, 0.25 for specimens one and two (20 ft. wide), 0.20 for specimens

three and four (16 ft. wide), and 0.80 for specimens five and six (40 ft. wide). The frequency

of these cycles was set in the actuator controller at 0.0833 Hz for test durations of 60 seconds.

Five cycles, or even possibly less, are adequate since this project does not incorporate the effects

of load fatigue.

While not a primary focus of this project, the last test of each specimen was an attempt to

cause failure. The CUREE protocol (Krawinkler et al. 2000) used for these tests is a deflection-

controlled quasi-static cyclic load history. This protocol is based on a finite series of cycles with

plateaus and peaks of increasing amplitude. Yield deflection, , was estimated for each


55/144


specimen and used as the reference deflection from which amplitudes of other cycles were

determined. If failure did not occur by the end of the series, the CUREE protocol allows for

additional cycles at higher amplitudes until the specimen fails.

2.7 TEST DATA ANALYSIS

As with any physical experiment, manipulation of raw test data (in this case, load and

deflection values) is required for a logical comparison of the different specimen configurations.

In this study, several specific variables are calculated in order to weigh the benefits and

detriments caused by changing test parameters as described in Section 2.4. These variables and

the methods used for their calculation are presented in the following sections.

Test data from each deflection and load measuring instrument was recorded by the DAQ

computer and entered into a spreadsheet format. Each column of data in the spreadsheet

corresponds to one of the twenty-four channels (instruments) being used, and is ordered

chronologically with time from the start of each test. A table listing each of these instruments,

its model and serial number, and calibration coefficient is in Appendix C.

These text-format spreadsheet files were later imported individually into a Microsoft

Excel calculation template. This template was programmed with the calculations necessary for

automatic computation of stiffness results and other important variables. The template also

provided instant load-deformation graphs for the data.

The first step of the calculations was to take the raw deflection data and convert it into

tared values by subtracting out the initial reading. For example, a string potentiometer with a

range of 10 in. is drawn out 5.25 in. and attached to the specimen. The first data entry for this

channel will indicate a deflection of 5.25 in. Therefore each of the data points in that column


56/144


must be tared by either subtracting or adding 5.25 in. depending on the direction of deflection in

order to attain the actual change in deflection.

Because the DAQ system had to be manually started and stopped in conjunction with the

independently controlled actuator, a section of data from the beginning and end of each test was

invalid. Therefore, the template was also programmed to shorten the data columns to include

only the meaningful data acquired during testing.

2.7.1 Yielding

While not a desired outcome of small-deformation stiffness testing, yielding is an

important concept to understand in terms of elastic versus plastic behavior. A material subjected

to a static load will undoubtedly undergo some deformation, though potentially immeasurably

small depending on its physical properties. If once unloaded, the material returns to its original

state, it is considered to have behaved elastically. However, if the material is loaded beyond its

elastic range causing permanent deformation even after being unloaded, then it has experienced

yielding. The force, fy, required to cause yielding is referred to as the yield point. Further

yielding caused by continued loading past this point, but before failure, is called plastic

deformation. Idealized elastoplastic response of a material subjected to force, f, causing

deformation, , is illustrated in Figure 2.22.


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Figure 2.22 Idealized Elastoplastic Force-Deformation Curve

force (f)

deformation ()

fy

y

failure

plastic behaviorelastic behavior


58/144


2.7.2 Global Deformation

Maximums and minimums were established from deflection or load readings of each

instrument. The rigid body motion determined from string potentiometers measuring slip in test

frame connections was subtracted from maximum positive and negative global deformations.

The resulting adjusted maximum global deformations were plotted against instrument location

distances along the length of the specimen. This curve represents a diaphragms shape at

maximum deformation, and also aids in visualizing effects of torsional irregularity. The sign

convention used for the purposes of this study is illustrated in Figure 2.23. Outward deformation

caused when the hydraulic actuator pushes out is considered positive. An actual diaphragm

deformation curve from Specimen 2, Test 8 (16 x 20 ft., no chords, with walls, corner opening,

blocked, nailed only) is presented in Figure 2.24. Note, the lop-sidedness (towards the right)

caused by a torsional irregularity due to the corner sheathing opening on that side.


59/144


Figure 2.23 Simplified Diaphragm Deformation Curve with Sign Convention

Figure 2.24 Diaphragm Deformation Curve - Specimen 2, Test 8

-

+

-

g

+

g

Distance AlongDiaphragm Edge

GlobalDeformation

(g)

0.230.19

-0.12

-0.16

-0.21-0.20

-0.15

0.13

0.19

0.24

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 2 4 6 8 10 12 14 16 18 20

Distance, L to R (ft)

GlobalDeformation,

RBMS

ubtracted

(in)


60/144


2.7.3 Cyclic Stiffness

Stiffness is the most basic and yet also the most useful of all the variables determined for

this study. In lay terms stiffness is simply a measurement of a structures capacity for resisting

deformation. Although stiffness can be expressed in several different forms, in general it is the

amount of force required to cause a known unit of elastic deformation. Thus, if a system is

linearly elastic, then its stiffness, k, can be described as the slope of the force-deformation curve

as illustrated in Figure 2.25.

Figure 2.25 Stiffness of a Linearly Elastic System

Cyclic loading requires a different approach to calculating stiffness. Cyclic loads change

the force-deformation plot from a straight line to an elliptical loop known as a hysteresis as

shown in Figure 2.26. Because the slope changes along the hysteresis, cyclic stiffness must be

approximated. A visual comparison of two common methods of determining cyclic stiffness

and their associated equations is presented in Figure 2.26. The peak-to-peak method shown in

Figure 2.26a approximates stiffness as the slope of an imaginary line between the points of

maximum positive and maximum negative deflection. Figure 2.26b shows the origin-to-peak

f

1

k


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method where stiffness is an average of slopes of imaginary lines from the origin out to the

points of maximum positive and negative deflection.


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(a) Peak-to-Peak Method

(b) Origin-to-Peak Method

Figure 2.26 Cyclic Stiffness Calculation Methods

f

max

+

max-

k+

+

+

+

=maxmax

maxmax ff

k

f

max

+

max-

k+

k-

2

+ +=

kkk

fmax+

fmax-


63/144


The spreadsheet program determined cyclic stiffness values using both the origin-to-

peak method and the peak-to-peak method. The first value calculated was cyclic stiffness

based on the force required from the actuator to cause a unit amount of deformation (excluding

rigid body motion) at the center of the diaphragm specimen. For tests in which the load-

deformation hysteresis is not centered on the origin due to uneven loading of the specimen, the

origin-to-peak method may not seem appropriate. While the peak-to-peak method may

seem better suited for such cases of uneven loading, a comparison of the two methods shows that

the cyclic stiffness results never varied by more than 3% in all 132 tests. Therefore, both

methods are assumed to be valid for test data exhibiting uneven loading. Due to the similar

results under either method and for brevity, this study will focus on analysis using the peak-to-

peak method from this point forward. (Note: the cause of uneven loading is assumed to be

residual stresses stored in the specimen from the previous test due to friction.)

As shown by Figure 2.27, either method produces similar results for a load hysteresis that

is well centered on the origin. Likewise, for hysteresis loops that are not centered on the origin

as shown by Figure 2.28, the two methods still yield similar results.


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-20

0

20

-0.25 0 0.25

Deformation (in)

Load(kips)

Figure 2.27 Cyclic Load-Deformation Hysteresis, Specimen 3, Test 3(Loops symmetric about the origin.)

-20

0

20

-0.25 0 0.25

Deformation (in)

Load(kips)

Figure 2.28 Cyclic Load-Deformation Hysteresis, Specimen 3, Test 24

(Loops not symmetric about the origin due to uneven loading.)

-9.45 kips

9.46 kips

-0.17 in

0.18 in

-15.18 kips

9.01 kips

0.15 in

-0.20 in

Origin-to-Peak:

kip/ft54.12

0.17-

9.45-

0.18

9.46

kcyclic =

+

=

Peak-to-Peak:

kip/ft0.5417.00.18

45.99.46k

cyclic=

+

+=

0.2% Difference of 0.1 kip/ft.

Origin-to-Peak:

kip/ft68.02

20.0

15.18-

15.0

01.9

kcyclic

=+

=

Peak-to-Peak:

kip/ft1.6920.00.15

18.159.01k

cyclic=

+

+=

1.6% Difference of 1.1 kip/ft.


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2.7.4 Shear Deformation

Global deformation is the sum of diaphragm shear deformation and flexural deformation

as illustrated in Figure 2.29. Since the load is applied at the center of diaphragm specimens, then

theoretically the shear deformation for each half of the diaphragm is equal. Diaphragm shear

deformation, visually detailed in Figure 2.30, can be determined from a geometric manipulation

of deflection results from the diagonal string potentiometers.

FLEXURAL

DEFORMATION

GLOBAL DEFORMATION

(Neglecting Rigid Body Motion)

SHEAR

DEFORMATION

(g)

(f) (s)

Figure 2.29 Diaphragm Deformation Theory


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F

F2

F2

bb

d

s

L2

L2

Diagonal String Pot.

(at zero deflection)

Diagonal String Pot.

(at deflection L)

Figure 2.30 Diaphragm Shear Deformation

First, the maximum diagonal deformation values in each direction for each half of the

diaphragm are determined, then averaged together, giving an average diagonal deformation, L,

for each half of the diaphragm, and for both positive and negative deformation. Using small

angle assumptions, shear deformation, s, can be expressed as:

2

Ls = (2.7.1)


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where is shear strain of the diaphragm and is calculated from the diagonal string potentiometer

deflections and geometric diaphragm properties by:

bd

dbL

22 +

= (2.7.2)

The data spreadsheet program calculated a s value for both halves of the specimen as

well as for maximum positive and negative diaphragm deformation. In some cases, shear

deformation for the left side of the diaphragm did not equal that of the right side and the positive

maximum did not equal the negative maximum. Though uneven loading can be attributed to

differences in positive versus negative shear deformation results, the differences in left side

versus right side shear deformation are due to different shear stiffness of each side.

2.7.5 Shear Stiffness

Shear stiffness is equal to shear force divided by shear deformation. Using the shear

deformations just determined, the spreadsheet program calculates shear stiffness for both sides of

the diaphragm using:

+

+

+

+=

SS

shear

VVk (2.7.3)

where V- and V+ are the maximum positive and negative shear forces applied to a side of the

diaphragm.

Under symmetric loading conditions and torsionally regular construction, the shear force

resisted by either side of the diaphragm is theoretically half of the load applied at the center by

the actuator. Thus, theoretically for torsionally regular test configurations, Vequals the reaction

force F/2 as shown by Figure 2.30, and can be verified by the load readings from the reaction

load cells at each side. However, asymmetric configurations such as a corner sheathing opening


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may cause the diaphragm to resist more of the actuator-applied load on the stiff side of the

diaphragm and less on the soft side. In such instances, Vshould be determined independently

for the left and right sides using the maximum readings from the left and right side load cells,

respectively. Accordingly, for torsionally irregular configurations (corner opening being the

only case for this study) the left and right side shear stiffness values must be kept separate.

As a caveat to this approach, a recurring problem during testing of Specimens 3, 4, and 5

was electrical malfunction of the side load cells, especially on the left side. An alternate method

by statics had to be used for tests in which there was reaction load cell malfunction. As

previously indicated, for a torsionally regular and symmetrically loaded specimen, each reaction

force equals half of the actuator-applied force. Similarly, for torsionally irregular specimens (i.e.

corner opening) the reaction forces combined, though not necessarily equal, should add up to the

actuator-applied force, F. If for example, the left side load cell malfunctions, its load can be

approximated by:

RL RFR = (2.7.4)

With the experimental shear stiffness, kshear, calculated for both the left and right sides of

the diaphragm specimen, the spreadsheet program uses elastic beam theory where:

s

sAG

LV

2

= (2.7.5)

where GAsis a more commonly accepted form of shear stiffness. Using the relationship:

s

shear

Vk

= (2.7.6)

Equation 2.7.5 may be solved for GAsin terms of kshearto give:

2

LkAG shears = (2.7.7)


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A shear stiffness value, GAs, is calculated for both the left and right sides of the

diaphragm. The left and right side values are averaged together for torsionally regular

specimens, but kept separate for torsionally irregular specimens in order to allow proper

comparison of results. For such cases it is possible that the shear stiffness on one side may be

considerably higher than the other.

2.7.6 Flexural Deformation

As shown visually by Figure 2.29, flexural deformation can be determined by:

sgf = (2.7.8)

The spreadsheet program averaged the maximum svalues for each half of the specimen for both

positive and negative deflection. These average shear deformation values were then subtracted

from the corresponding maximum positive and negative global deformations to give a maximum

positive and a maximum negative flexural deformation.

2.7.7 Flexural Stiffness

Experimental flexural stiffness, kf, may be expressed as:

+

+

+

+=

ff

f

FFk (2.7.9)

The spreadsheet uses the above peak-to-peak equation in order to arrive at one flexural stiffness

value.

Elastic beam theory offers an approach to a more common form of flexural stiffness.

Theoretical flexural deformation of a beam with a concentrated load applied at the center is:

EI

FLf

48

3

= (2.7.10)


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where EI is the flexural stiffness of the beam, or, as in this case, the diaphragm. Solving

Equation 2.7.10 forEIin terms of kf (recalling that kf = F / f) gives:

48

3LkEI

f= (2.7.11)

The spreadsheet program calculated flexural stiffness in the above form for each diaphragm test.

2.7.8 Hysteretic Energy

Hysteretic energy is the energy dissipated during one cyclic loading of a structure and

may be quantified as the area inside a load-deformation hysteresis for one cycle. The area within

a hysteresis from an experimental test can be approximated by numerical integration. Numerical

integration involves averaging the load values of two consecutive points along the curve.

Multiplying this average load by the difference between deformation values of the same two

consecutive data points gives the area under the curve between those two data points.

Geometrically, the calculation equates to determining the area of very narrow trapezoid. This

process must be repeated for every pair of consecutive data points all the way around the loop.

Depending on the location along the curve with respect to the deformation axis, the area is

considered either positive or negative. The net total of these incremental areas is the hysteretic

energy as represented algebraically in Equation 2.7.12:

( ) ++

+

+=

n

n

nnnnd

ffE

1

11

2 (2.7.12)

where the integer nrepresents the individual data points around the entire loop. The spreadsheet

program follows the same process described above to arrive at a value for hysteretic energy for

each diaphragm test. However, for greater accuracy, the program calculates a total area for three

consecutive loops around the hysteresis (using data points only from the middle three loops) and


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then divides the total by three, yielding an average area inside only one loop. See Figure 2.31 for

a visual description of numerical integration of a hysteresis.

Figure 2.31 Numerical Integration of a Load-Deformation Hysteresis

f

a

Area under curve between points

aandbto be considered positivefa

b

fb

+

a

b

f

c

fc

d

fdd

c

-

Area under curve between points

canddto be considered negative


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2.7.9 Equivalent Viscous Damping

Damping is the mechanism that causes gradual reduction of vibration in a system and

thereby a loss of energy. In structures, damping and the resulting energy loss is caused by a

variety of conditions such as internal friction of materials subjected to repeated deformations,

friction from movements at connections, opening and closing of cracks, and friction with

external or nonstructural systems with which the structure is in contact. Since the systems that

can cause damping are seemingly limitless and difficult to identify, a mathematical model

capable of predicting actual damping is nearly impossible.

Thus, a concept called equivalent viscous damping is used to represent all of the damping

mechanisms for a simplified approach under the assumption that the structure behaves as a

Kelvin solid viscoelastic element (Fischer and Filiatrault 2000). Equivalent viscous damping is

defined by equating the energy loss during a vibration cycle (i.e. the hysteretic energy as defined

in Section 2.7.8 and graphically as shown in Figure 2.32) in an actual structure to that of an

equivalent viscous system.


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Figure 2.32 Damping relationship to an equivalent viscous system

Using Figure 2.32 and setting the hysteretic energy, Ed, from one experimental cycle

equal to that of the equivalent viscous system gives:

22 oo

neqd XkE

= (2.7.13)

where eqis the equivalent viscous damping ratio, is the experimental test frequency, nis the

natural frequency of the test structure. Strain energy,ESo, is equal to the area of triangle OAB

(or triangle OCD) from Figure 2.32 and can be calculated from experimental stiffness, ko at

maximum deformationXo:

2

2

oo

OABSo

XkAE == (2.7.14)

Substituting Equation 2.7.14 into Equation 2.7.13 gives:

OABn

eqd AE 4= (2.7.15)

f

Xo

+

Xo-

ko+

ko-

B

A

OD

CEd = Hysteretic Energy(Area enclosed by hysteresis loop)


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Although for this experiment does not actually equal n, for simplicity the assumption as such

does allow for an acceptable approximation of eq (Chopra 1995). Assuming = n and

solving for the equivalent viscous damping ratio gives:

OAB

deq

A

E

4= (2.7.16)

As shown in Figure 2.32, the maximum deformations in both the positive and negative directions

are not necessarily equal. Therefore, the equivalent viscous damping ratios should be calculated

for triangle OAB using ko+ and triangle OCD using ko

- and then averaged. Thus, equivalent

viscous damping ratio, eq, is commonly expressed as:

So

deq

E

E

4= (2.7.17)

whereESois strain energy (equal to the area of triangle OAB or OCD).

For the purposes of this study, the spreadsheet program calculates the areas on both the positive

and negative sides of the curve using cyclic stiffness values determined by the origin-to-peak

method. The spreadsheet takes an average of the two strain energy values and the hysteretic

energy previously calculated, and uses a formula based on Equation 2.7.17 to determine the

equivalent viscous damping ratio for each diaphragm test.

Although equivalent viscous damping is not technically correct for the tests in this study

due to some non-linearity in the load-deformation response, the maximum specimen deflections

were kept low to minimize error. The viscous damping term is a measure of all of the damping

in the system (hysteretic and material) and is used for modeling the more complex system as a

simplified mass-spring-dash pot system. The equivalent viscous damping is the value for the

dash pot. While the design community views the concept of equivalent viscous damping as very


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inaccurate for calculating damping in an actual structure, it is used in research simply as a means

of comparing damping capability.


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CHAPTER III: RESULTS AND DISCUSSION 67

CHAPTER III

RESULTS AND DISCUSSION

3.1 INTRODUCTION

The objective of this diaphragm study was to test the stiffness of wood diaphragm

specimens under varying configurations in order to develop a method for determining shear

stiffness similar to that already used by the cold formed steel industry. Combined, 132 non-

destructive stiffness tests were performed on 6 different specimens. Due to the high volume of

data produced, a test-by-test analysis and comparison of results would be monotonous and not in

the best interest of the reader. Therefore, the calculations applied to each set of test data will be

thoroughly explained step-by-step in general terms. The remainder of this chapter will be

devoted to discussion of trends in test results and relating those trends back to the various

construction parameters. The individual test results are presented in Appendix A.

3.2 TEST CONDITIONS

As a foreword to discussion of results, the reader needs to be aware of the ever-changing

conditions encountered during testing, namely weather. Testing of diaphragm specimens began

on January 12, 2001 and continued until July 6, 2001. Throughout those six months the weather

played a substantial role in the test schedule, specimen moisture content, and periodic

malfunction of instruments and equipment.


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CHAPTER III: RESULTS AND DISCUSSION 68

As indicated in Chapter 2, the diaphragm testing facility is an outside concrete slab with

no protection from the weather. A large tarp was always used to cover speci

bott masters thesis etd

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