Citation
A study of metal deck diaphragm participation in the St. Louis downtown convention and stadium facility roof

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Title:
A study of metal deck diaphragm participation in the St. Louis downtown convention and stadium facility roof
Creator:
Petersen, Jack Edward
Publication Date:
Language:
English
Physical Description:
xiv, 338 leaves : illustrations (some folded) ; 29 cm

Subjects

Subjects / Keywords:
Convention facilities -- Design and construction -- Missouri -- Saint Louis ( lcsh )
Stadiums -- Design and construction -- Missouri -- Saint Louis ( lcsh )
Roofs -- Design and construction -- Missouri -- Saint Louis ( lcsh )
Convention facilities -- Design and construction ( fast )
Roofs -- Design and construction ( fast )
Stadiums -- Design and construction ( fast )
Missouri -- Saint Louis ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references.
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Civil Engineering.
General Note:
Department of Civil Engineering
Statement of Responsibility:
by Jack Edward Petersen.

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Auraria Library
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Auraria Library
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
28141474 ( OCLC )
ocm28141474
Classification:
LD1190.E53 1992m .P47 ( lcc )

Full Text
A STUDY OF METAL DECK DIAPHRAGM PARTICIPATION
IN THE ST. LOUIS DOWNTOWN CONVENTION AND STADIUM
FACILITY ROOF
by
Jack Edward Petersen
B.S., University of Colorado, Boulder, 1983
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Civil Engineering
1992
j-.i
i


This thesis for the Master of Science
degree by
Jack Edward Petersen
has been approved for the
Department of
Civil Engineering
by
Andreas Vlahinos
Ernest Harris
&!o y
Date


Petersen, Jack Edward (M.S., Civil Engineering)
A Study of Metal Deck Diaphragm Participation in the St.
Louis Downtown Convention and Stadium Facility Roof
Thesis directed by Professor Judith J. Stalnaker.
ABSTRACT
The new St. Louis Downtown Convention and Stadium
Facility is a covered multi-use facility, with a clearspan
roof covering an area 600'x730'. In the Design Office the
structure was analyzed and designed based on a bare frame
model considering only primary trusses. Independent
studies of diaphragm behavior, thermal restraint, and other
issues were conducted by approximate analysis methods. To
address numerous uncertainties associated with these
analyses, an analysis considering the full structure
including secondary framing and the metal deck diaphragm
was undertaken.
The primary computer model studied included all
trusses, joists, bracing, and the diaphragm under gravity
loadings. The metal deck diaphragm was modeled as a thin
flat plate using four node membrane elements. The
thickness of these elements was computed to match the
stiffness of the actual metal deck as predicted by Steel


2.2.5 Material Properties .................. 39
2.2.6 Deflection Limitations.................40
2.3 Design Office Analysis ...................... 40
2.3.1 Schematic Analysis Model...............43
2.3.2 Design Office Model .................. 43
2.3.3 Studies Involving Metal Deck
Diaphragm..............................47
2.3.3.1 Evaluation of Thermal
Restraint.......................48
2.3.3.2 Tension Ring/Shell
Action..........................49
2.3.3.3 Deck Warping...................53
2.3.3.4 Approximation of Deck
Lateral Stiffness...............54
2.3.3.5 Evaluation of Drag
Strut...........................54
2.4 Project Analysis Model ...................... 55
2.4.1 Description of Project
Analysis Model.........................56
2.4.1.1 Model Geometry ............... 57
2.4.1.2 Member Properties..............58
2.4.1.3 Metal Deck Diaphragm ......... 61
2.4.1.4 Development of
Membrane Element
Properties......................63
2.4.1.5 Development of Finite
Element Mesh....................69
vn


2.4.2 Additional Analysis Models.............70
3. Evaluation of Analysis Results......................74
3.1 Evaluation Criteria...........................77
3.2 Gravity Load Project Analysis
Model.......................................8 0
3.2.1 Deflection.............................81
3.2.2 Member Forces..........................88
3.2.3 Diaphragm Shears.......................93
3.2.3.1 Typical Bay Area................97
3.2.3.2 North and South Ends...........103
3.2.3.3 Corner Areas ................. 105
3.3 Additional Models............................112
3.3.1 Gravity Load Model Relief
Joint.................................112
3.3.2 Gravity Load Model with
Tension Ring Joint....................114
3.3.3 Lateral Analysis Models ............. 118
3.3.4 Combined Forces.......................122
4. Summary and Conclusions............................125
4.1 Recommendations..............................128
4.2 Topics for Additional Study..................132
Appendix
A. Figures of Construction Drawings for the
St. Louis Downtown Convention and Stadium
Facility..........................................139
viii


B. Hand Calculations from Approximate Design
Office Analyses Regarding the Metal Deck
Roof Diaphragm.....................................167
C. Hand Calculations Supporting the
Development of the Project Analysis
Model..............................................194
D. Listing of Project Analysis Model
Computer Files.....................................216
D1 SAP80 Input File Listing..............217
D2 SAP80 Joint Displacement,
Applied Force and Reaction
Output File.............................232
D3 SAP80 Membrane Force Output
File (partial)..........................249
Bibliography .......................................... 337
IX


FIGURES
Chapter 1
1-1 Architect's Rendering of The St. Louis
Downtown Convention and Stadium
Facility.......................................... 2
1-2 Design Office Graphics Plot Plan View............ 8
1- 3 Typical Planar Diaphragm...........................15
Chapter 2
2- 1 Section Through Typical Concrete Bent ............ 22
2-2 Unbalanced Snow Load Profiles......................36
2-3 Summary of Wind Pressures..........................38
2-4 Plan View Key Map of Design Office Model...........42
2-5 Graphics Plot, Schematic Model.....................44
2-6 Graphics Plot Design Office Model
Perspective View..................................4 6
2-7 Response of Metal Deck Subject to
Differential Thermal Change ....................... 50
2-8 Key Map of Project Analysis Model
(Oversized) ............................ Rear Pocket
2-9 Typical Cantilevered Diaphragm.....................64
2-10 Typical Load/Deformation Curve for Metal
Deck Diaphragm......................................67
2-11 Finite Element Mesh for Initial Project
Analysis Model (oversized)................Rear Pocket
2-12 Partial Key Map for Project Model with
Relief Joints File SL6..............................73
x


Chapter 3
3-1 Summary of Analysis Results Project
Analysis Model (oversized)................Rear Pocket
3-2 Horizontal Deflected Position at Slide
Bearings Due to Gravity Loads.....................86
3-3 Vertical Displacements of Primary Trusses
Due to Gravity Loads................................87
3-4 Metal Deck/Diaphragm Bracing Interaction...........89
3-5 Locations of Overstress in Metal Deck
Diaphragm Due to Superimposed Dead +
Live Load...........................................94
3-6 Deck Warping at North End..........................96
3-7 Typical Bay Study Summary..........................99
3-8 Forces at Tension Ring Joint Corner
Areas..............................................108
3-9 Plan View of Deck Relief Joint....................113
3-10 Summary of Diaphragm Overstress at Corner
Interface Area First Additional Model .......... 115
3-11 Summary of Analysis Results Project
Analysis Model with Jointed Tension Ring
(oversized) .............................. Rear Pocket
3-12 Summary of Analysis Results Lateral
Model with Tension Ring Intact
(oversized) .............................. Rear Pocket
3-13 Summary of Diaphragm Overstress Lateral
Load Plus Dead Load on Plan........................121
3-14 Summary of Diaphragm Overstress Lateral
Load Plus Dead Load on Project Analysis
Model....................................................124
xi


Appendix A
A-l General Notes Defining Material
Properties.......................................14 0
A-2 1/64" Scale Plan Bottom Chord
Framing...........................................141
A-3 1/32" Scale Plan Bottom Chord
Framing Quadrant 'A'..............................142
A-4 1/64" Scale Plan Top Chord Framing...............143
A-5 1/32" Scale Plan Top Chord Framing,
Quadrant 'A'......................................144
A-6 1/64" Scale Primary Truss Elevations...............145
A-7 Typical Specification of Parameters for
Prefabricated Joists..............................146
A-8 Typical Built-Up Top Chord Details..................147
A-9 Joist Girder/Truss Connection ..................... 148
A-10 Typical Bracing Connection.........................149
A11 Top/Bottom Chord Connection at Bearing.............150
A-12 Typical Bottom Chord Truss Intersection .......... 151
A-13 Truss Top Chord Panel Point Connection.............152
A-14 Truss Top Chord Panel Point ...................... 153
A-15 Intersection of Primary Trusses at Top
Chord.............................................154
A-16 Intersection of Primary Trusses at Top
Chord.............................................155
A-17 Typical Joist Girder Profile.......................156
A-18 Typical Joist Girder to Truss Connection...........157
A-19 Diaphragm Bracing System...........................158
Xll


A-20 Plan at Typical Truss Slide Bearing...............159
A-21 Section at Typical Truss Slide Bearing............160
A-22 Typical Slide Bearing at Lateral
Restraint..........................................161
A-23 Typical Perimeter Detail..........................162
A-24 Typical Deck/Chord Connection ................... 163
A-25 Expansion Joint at Tension Ring...................164
A-26 Expansion Joint at Tension Ring...................165
A-27 Typical Metal Deck Relief Joint ................. 166
xm


TABLES
Chapter 2
2- 1 Summary of Analysis models.........................72
Chapter 3
3- 1 Summary of Gravity Load Reactions at
Bearings............................................75
3-2 Lateral Displacements at Slide Bearings
Due to Dead Loads...................................82
3-3 Vertical Displacements at Primary Truss
Joints Due to Dead Loads............................84
3-4 Vertical Displacements of Joists at Corner
Areas Due to Dead Loads............................107
3-5 Lateral Displacement at Slide Bearings
Due to Lateral Loads...............................120
xiv


l. Introduction
The purpose of this project is to study the
participation of a metal deck diaphragm in stiffening a
long-span roof structure. The project was undertaken to
identify detrimental effects associated with the metal
deck participation, such that the structure may be
detailed appropriately. This study is specific to one
structure, the new St. Louis Downtown Convention and
Stadium Facility (SLDCSF), which has a "domed"-shaped
roof spanning 600' x 730' (see Figure 1-1).
The stiffening effect of metal deck diaphragms in
typical buildings is well documented in technical
literature and design methods outlined in the Steel Deck
Institute's (SDI) publications (Luttrell, 1987). Prior
to 1960 most steel frame buildings included structural
steel bracing to adequately brace compression flanges of
beams and to resist lateral loads. Although the
stiffening and strength of metal deck diaphragms was
recognized, it was considered a beneficial redundant
system due to a lack of established design criteria.
Research supported initially by individual decking
manufacturers and later by trade organizations such as
1


sjaja
Figure 1-1 Architects Rendering of The St. Louis Downtown Convention and
Stadium Facility
2


SDI, resulted in the first comprehensive design manual
for metal deck diaphragms in 1981 (Luttrell, 1981a). All
published research reviewed in the author's literature
search, as well as the SDI design recommendations, are
directed towards planar diaphragm strength and stiffness.
This paper will discuss assumptions and analyses
used in two separate evaluations of the SLDCSF. The
first is a review of the analysis done in a consulting
engineering office in which the metal deck diaphragm was
neglected. The entire structure was analyzed with steel
elastic beams elements, and bracing diagonals were added
as required to maintain stability. This first analysis
model is referred to as the "Design Office Model"
throughout the balance of this paper. A second model was
constructed to evaluate the participation of metal
decking on the structure. In this model, hereinafter
referred to as the "Project Analysis Model", a finite
element membrane was added to the framed structure to
model the shear stiffness of the diaphragm. The Project
Analysis Model represents a level of detail considerably
above normal practice in consulting engineering of
building structures. It was constructed to answer
questions identified in the first model, and offers an
3


interesting secondary theme for this paper: Does the
Design Office Model adequately describe the structure's
behavior?
1.1 Design Office Model
The structural analysis of the SLDCSF in the Design
Office Model did not include the metal deck diaphragm or
any stiffness or equivalent members to simulate the
diaphragm's behavior. In design, the primary trusses are
braced by, and all lateral load is resisted by,
structural steel bracing (see Figure A-19 in Appendix A)
in the plane of the diaphragm. Although this design
assumption is consistent with design practice prior to
1960, the diaphragm was neglected not for lack of design
rules, but due to the following factors:
Truss Chord Bracing Requirements:
Primary trusses on SLDSCF have top chord weights in
excess of 800 pounds per foot and typically carry a
3,000 kilo pound (kip) axial load (Figure A-8). The
"rule of thumb" bracing load of 2% of the axial
force, would require the decking to carry 60 kips
over some small, and by code undefined length of
metal deck. Typical deck fastenings are inadequate
4


to resist this magnitude of force. A 12 1/2" gap
(see Figure A-9) was required between metal deck
bearing and the top of the trusses to accommodate
detailing of girder-to-truss connections. The
calculated bracing force would have to have been
carried through this gap, presenting additional
difficulties.
Lateral Loads
Lateral loads due to earthquake motions on the
structure result in deck shears in excess of 2,000
pounds per foot. This is well in excess of SDI
allowable diaphragm capacity for the deck selected
based on gravity load requirements. Commercially
available metal deck has a maximum diaphragm
capacity of about 700 pounds per foot in the span
range found on the SLDCSF.
Lower Bound Assumptions
By neglecting the strength and stiffness of the
diaphragm it was felt that the design would be
conservative, and would provide additional reserve
capacity.
5


This assumption of providing a 'braced' roof system
also resulted in numerous uncertainties which led to this
study. These questions center on identifying any
disadvantages to performance of the structure as a result
of the metal deck carrying load. Since the primary
'braced' system is designed to carry all loads
identified, participation of the deck is not a structural
safety issue (with exception of a possible undesirable
redistribution of seismic forces). Problems associated
with a deck carrying load in excess of its ultimate
capacity are primarily issues of serviceability. Shear
in the metal deck diaphragm is induced when the
supporting structure undergoes unbalanced displacement
after the deck is fastened. Areas which overstress
connections in shear may result in a total loss of future
wind uplift capacity. Buckling or distortion of areas of
deck may be detrimental to insulation and roof membranes,
may be visually unacceptable and may result in compromise
of the structures waterproofing. Finally, in the event
that the deck system is too stiff, response to seismic
dynamic load may be altered from that analyzed.
6


1.2 Project Analysis Model
The Project Analysis Model was constructed to
identify areas of the roof deck system which may present
serviceability problems under design loads, and then
develop appropriate relief joints and details to
alleviate overstress. The following conditions were
identified as potential serviceability problems:
1. Tension Rinq/Shell Action
In the Design Office Model, the two-way primary
trusses, diaphragm bracing and bottom chord
bracing were included in the analysis model
(see Figure 1-2). The actual structure (see
Figure A-5) includes a continuous heavy wide
flange section (tension ring) that runs around
the perimeter of the dome and is required to
support gravity load. The metal deck diaphragm
(which is stiffened by steel joists at eight to
12 feet o.c.) is not included in the model.
These elements, to some degree, begin to act as
a shell reacting against the tension ring.
This structure is atypical for diaphragm design
since due to the "dome" shape, the metal deck
will attempt to resist both gravity and lateral
7


Figure 1-2 Design Office Graphics Plot Plan View
8


forces. Since the structure is built of
faceted panels, as opposed to one planar
diaphragm on which SDI criteria is based,
reacting forces at boundaries (joists and
trusses) must be carefully evaluated. The
study of this behavior becomes a matching of
strength and stiffness, identifying areas which
are too stiff (thus attracting too great a
load) for the code-specified allowable
strength.
2. Thermal Motions/Restraint
During the erection period of a roof of this
size it is possible the steel structure may
undergo temperature changes in excess of 50F
from service level. During erection, large
areas of galvanized deck may be exposed to
direct sunlight. Due to the difference in
thermal mass and exposure conditions (decking
already applied will shade trusses), a large
differential temperature between deck and
primary trusses is possible. Such
differentials give rise to restraining forces
at the diaphragm/ structural steel interface.
9


3. Lateral Stiffness
The roof structure of the SLDCSF bears on eight
sections of concrete structure which are
isolated by expansion joints and are laterally
independent (refer to Figure A-3). To
determine the appropriate distribution of
seismic forces, a dynamic analysis is to be
performed on the system for final design. The
stiffening effect of metal deck may
significantly alter the mass/stiffness ratio of
the roof system, thus the distribution of load
within the full structure model.
Another potential problem area is
diaphragm shear transfer to the ring beam at
the perimeter. The ring beam acts as a
collector to deliver lateral seismic force from
the structure below and to resist applied wind
force. As detailed, all load is delivered at
discrete points where diaphragm bracing is
connected to the ring beam (see Figure A-10).

10


If the metal deck has adequate strength, it
will load the perimeter beam along the entire
length of the structure, thus introducing drag
forces along a greater distance.
4. Detailing for Deflections
The roof structure has the potential to move
through very large displacements relative to
the cast-in-place structure on which it bears
(See Section 2.1.6). Accommodation of these
displacements requires costly bearings and
waterproofing details. If, due to the
stiffening effect of the deck, smaller
displacements could be designed for,
significant cost savings could be realized.
5. Erection Conditions
In the Design Office Model, the structure was
assumed "fully shored" and the order in which
the structure is built is of no consequence.
In long-span roofs, particularly those
featuring two-way designs, the sequence of
shoring and erection is critical to the final
design forces. One scenario being discussed
for erection of the SLDCSF is to use two
11


shoring towers to build two trusses, and
install all joists, decking, and bracing. The
shores are then released, moved, and reused on
the next truss. This type of sequence results
in potentially forcing a completed diaphragm
through unequal displacements at each end (as
shoring is removed) and inducing shear in the
plane of the diaphragm.
This behavior was studied by the author
briefly in preparation of the contract
documents for the Fargodome, a covered stadium
in Fargo, North Dakota. To reduce anticipated
problems, relief joints were used in deck, and
deck was installed when much of.the roof dead
load was already in place. No unacceptable
conditions (i.e., connector failure, buckling,
etc.) were observed during the construction
period, however, very little live load (snow)
was applied at this time.
The study of the SLDCSF roof, using the Project
Analysis Model, focused on items 1 and 3 listed above.
Item 2 has been identified and studied by the author on a
number of projects. A brief summary of solutions
12


typically used and a study of conditions unique to the
SLDCSF are included. Item 4 is an engineering judgment
item of which results of the Project Analysis Model are
invaluable for guidance. Finally, item 5 is not
addressed in this paper as the final erection scheme for
the structure was not developed at the time of writing.
1.3 Introduction to Metal Deck Diaphragm Terminology
The term "metal deck diaphragm" is used to describe
corrugated metal sheets and their supporting members
connected in such a manner to resist forces in the plane
of the decking. Such diaphragms behave in a manner
similar to deep, short span plate girders within their
range of allowable loads. A simple, planar, rectangular
diaphragm is shown in Figure 1-3 to define the basic
elements common to all diaphragms.
The diaphragm consists of thin metal sheets which
have been rolled into a corrugated pattern to improve its
bending strength normal to the flat surface.
Commercially available decks range in thickness from 12
gauge (0.1046") to 26 gauge (0.0179") with a rolled depth
of 0.6" to 3", and sheet widths of 24" to 36". The
decking is connected to supporting members at the point
13


of contact with the joist (or so called "lowhat") via
welding, self-tapping screws or steel drive pins. The
sheet-to-sheet connections ("sidelaps") are typically
connected with self-tapping screws or welding. The beams
at the perimeter of the diaphragm act as chords analogous
to flanges of a plate girder, and beam to beam
connections at the chord must be carefully detailed in
large diaphragms to maintain continuity.
As seen in Figure 1-3, the diaphragm acts to resist
shear by carrying the applied shear force (P) through the
metal decking, and delivering the bending or overturning
forces (RoT) to the chords. Strength of diaphragm is
influenced by decking thickness, connector strength and
pattern, deck span and other factors. Metal deck
diaphragms are typically considerably more flexible than
a steel sheet of the same thickness due to flexibility in
the corrugations, slip of fasteners, and other factors.
The aspect ratio of the diaphragm (a/L in Figure 1-3),
deck span and connector pattern also influence the
deflection (A). In the publication Diaphragm Design
Manual. (Luttrell, 1987) rational design equations to
predict diaphragm strength and stiffness are developed.
14


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1.4 Development of Design Methods for Metal Deck
Diaphragms
The stiffening effect of corrugated metal sheets
(metal decking) adequately fastened to a structural steel
frame has been recognized for many years. The use of
metal deck as a diaphragm may provide adequate lateral
support for the compression flange of small steel beams
and joists, and can serve to transfer wind or earthquake
forces between vertical elements of the lateral system of
a structure. Despite this observation, most steel
building structures designed prior to 1960 neglected the
beneficial diaphragm action and included structural steel
bracing for lateral loads.
In the 1950's research directed at quantifying the
behavior and strength of corrugated metal diaphragms
began. Nilson (1960) provided the first significant
summary of strength and flexibility of corrugated shear
panels. His paper summarized five years of research done
at Cornell University on 10 different deck profiles.
Welded connections were studied in detail and
recommendations on factors of safety made. Bryan and El-
16


Dakhakhni (1968) studies formed the foundation of all
critical parameters defined in the current SDI design
formulas.
Their work studied the stiffness of shear diaphragms
based on various fastening configurations, including
decks which were connected at intermittent flutes to the
supporting steel. Flexibility calculations by these
authors include warping of corrugations, geometry of
corrugations, axial stiffness of supporting steel,
stiffness of the chords (margin) of the diaphragm, and
slip in sidelap and primary fasteners. Study of
fasteners included strength and stiffness of self-tapping
screws used in deck-to-purlin connections, and rivets for
side lap connections. Finally, this study included
calculated and full scale testing of purlin-to-primary
steel connections for both strength and stiffness. This
study demonstrated that deformation of the purlin between
the diaphragm and steel typically accounts for at least
50% of the overall flexibility of the diaphragm system.
Current SDI publications neglect this component of
deformation and no mention is made in the commentary.
Recent concern expressed by engineers has resulted in new
studies of "rollover" behavior of light, double-angle
17


seats on K-series open web steel joints when subjected to
shear transfer perpendicular to the direction of the
joist span. Review of a draft copy by Fischer, West, Van
de Pas (1991) describes a limited number of test results
made by Vulcraft, Inc., a major steel joist producer.
Easley (1975) identified and studied buckling of
corrugated shear diaphragms, similar to tension field
action in plate girders. Buckling was shown to control
the strength of only thin diaphragms with strong, closely
spaced connections. In roof decks which are readily
available commercially (types A, B, F, and N per SDI),
buckling rarely controls, but the equations have been
incorporated into SDI Diaphragm Design Manual (Luttrell,
1987) in a modified form.
Additional refinements to Bryan's work were made by
Davies (1976, 1977), who identified the controlling
mechanism of corner fasteners. Davies also studied the
use of orthotopic finite elements to analyze corrugated
metal shear panels and recommended an appropriate
diagonal term. No additional literature on this idea
(which conflicts with earlier testing by Nilson) has been
published since this time.
18


In 1981, the Steel Deck Institute published the
first edition of the Diaphragm Design Manual (Luttrell,
1981a). This document includes generic equations for
predicting flexibility and strength of metal deck formed
in any profile. It accounts for all sources of
deformation identified by earlier researchers in
analytical terms. Finally, this manual defines safety
factors and design methods for typical planar diaphragms
and their connections. Published with the diaphragm
manual was Steel Diaphragm Studies (Luttrell and Huang,
1981) which described the testing program and code
development in detail.
SDI published a second edition of the Diaphragm
Design Manual (Luttrell, 1987) which incorporated
additional information on fasteners and design examples.
Since this time, no additional information was found in
the literature. Dr. Luttrell (1992), SDI's technical
adviser, noted in a phone conversation that little
additional research has occurred since a rational design
method was presented. Most testing since 1980 has been
done for and supported by individual manufacturers to
obtain model building code agency (ICBO, BOCA, SBC, etc.)
approvals.
19


No literature was discovered regarding the
participation level of diaphragms on actual structures.
A few articles on the stiffening effect of curtain walls
on structural building frame were reviewed; however,
these were not applicable to metal deck diaphragms.
20


2. Description of Analysis Models
The roof of the SLDSCF is a very complex structure.
Although the basic structure and its analyses were
described briefly in the first chapter, it is necessary
to have a greater knowledge of details of the structure
to interpret the results. This chapter is provided to
familiarize the reader with details of the structure.
Methods of analysis used and assumptions made in building
each of the analysis models are defined. For the
reader's convenience in cross-referencing, all plans and
detailed figures of the structure are included in
Appendix A.
2.1 Structure Description
The structure upon which this study was based is a
new convention/stadium facility for the City of St.
Louis. The finished structure is planned to have 700,000
square feet (S.F.) and seat 70,000 people for a stadium
event. Although this paper's focus is upon structural
behavior of only the roof system, a brief description of
the substructure is included for understanding of the
roof bearing conditions. The substructure consists of a
21


cast-in-place frame/shear wall system with precast double
tee floor framing system and cast-in-place topping. The
overall dimension of the structure is approximately 730'
x 600', reguiring eight expansion joints in the structure
below the roof to accommodate shrinkage and thermal
displacements. The structure has five levels above
grade, resulting in a roof bearing elevation of 138'-8
above ground. A schematic typical section through the
substructure is shown in Figure 2-1.
The roof structure of the SLDSCF is supported only
at the perimeter columns, resulting in an elliptical plan
shape of approximately 405,800 S.F. (refer to Figure A-
4). The roof structure rises towards the center
following the profile of a parabolic arch in the shorter
(600') direction. The roof structure contains no
t .
permanent expansion joints since the primary trusses
spanning in each direction must bear at far ends of the
stadium. This results in a single unit roof effectively
bearing on eight separate structures below (due to
expansion joints) which move independently due to thermal
changes and under lateral (wind and seismic) load.
22


Figure 2-1 Section Through Typical Concrete Bent
23


The roof structural system for the SLDCSF studied in
this paper is a two-way structural steel truss with pre-
fabricated open web steel joists and joist girders used
as secondary framing. Figure A-5 is a 1/32" scale
quadrant plan of the roof framing at the point in design
when the computer model to study diaphragm behavior was
started. This was at approximately 50% completion of
construction documents (Design Development stage).
Subtle changes have continued to be made to the structure
as design has evolved; however, this point was selected
as a baseline for comparison.
2.1.1 Selection of Structural System
The selection of structural system for the roof
resulted from a lengthy value engineering study. The
most desirable system aesthetically was a two-way space
truss with cable bottom chords and a fabric (teflon-
coated, high-tensile fiberglass) roof. This system was
engineered to a level where quantities could be
accurately defined and proved too costly. Six additional
schemes were engineered to a similar level and priced.
Pricing considerations included total weight and cost of
24


materials, required shoring, erection difficulty, and
impact on construction schedule of the remainder of the
project (substructure, cladding, finishes, etc.)*
The scheme selected (Figure A-5) proved least costly
due to relative simplicity, high quantities of
prefabricated components, and low unit costs of structure
steel (wide flange shapes and pipe). It is desirable
structurally due to the two-way action which provides
redundant load paths and a higher margin of safety
against collapse than some of the one-way systems
considered. Finally, the system met the architectural
criteria to "frame" the football playing field with a
primary truss on each side.
2.1.2 Primary Truss Framing
The two-way system is made up of five primary
trusses in the short (600') direction, and two primary
trusses in the long (730') direction. The primary system
consists of planar structural steel trusses, 65' deep at
their deepest point (centerline). Configurations of each
of the primary truss types is shown in Figure A-6. The
top chord of the primary trusses is constructed of two
36" deep wide flange sections, with webs vertical, spaced
25


30" apart. The two members are made composite by virtue
of continuous plates on the top and bottom flange and
intermittent diaphragm plates to carry shear (see Figure
A8). This configuration was necessary due to the very
long, unbraced lengths of 65' at the top chord. Metal
deck was inadequate in strength to brace the top chord
between panel points.
The bottom chords of the primary trusses are
detailed as a single W14 wide flange shape (see Figure A-
12). The roof experiences no net uplift due to wind
forces based on wind tunnel testing (or other load
cases), hence stability of these members in compression
was not a design parameter. The vertical members in the
truss are 16" diameter and 18" diameter pipes. Pipe
sections were selected due to their suitability to long,
unbraced lengths and the relatively light compression
loads to be resisted. Finally, the diagonals of the
primary trusses are constructed of high-strength
structural strand (cable). A typical joint showing the
connection of pipes and cables at bottom chord is shown
in Figure A-12. Under balanced loads, the trusses behave
essentially as a tied arch, applying virtually no load to
the web members of the truss. For unbalanced loadings
26


and vertical bracing of the top chord, these diagonals
carry small forces. The web system was carefully
evaluated during design to assure adequate strength and
stiffness to resist arch type buckling of the top chord.
Details of the primary truss system are shown in Figures
A-13 through A-16.
The overall span-to-depth ratio of the primary
system is 11.25:1 for the longer trusses, and 9.25:1 for
the shorter trusses. Due to the very large depth, it was
assumed in design that all pieces would be shipped loose
and assembled in the field. Accordingly, connections
were detailed using high-strength bolts wherever
possible, with the majority of welding done in the shop.
2.1.3 Secondary Framing
Secondary framing between major trusses is designed
using open web steel joists and joist girders to the
greatest degree possible. These trusses are
prefabricated in the joist manufacturer's shop for
loadings and geometries specified on the structural
drawings. See Figure A-7 for example of specification of
27


loads. Joists were selected due to low unit costs as
compared with structural steel custom trusses, and since
they require a minimum of preassembly on the site.
Joist girders were used to frame between panel
points of the primary trusses (refer to Figure A-17). In
addition to supporting joist framing running up and down
the slope, these members were framed at top and bottom
chord to provide bracing for primary trusses (refer to
Figure A-18). The joist girder top chords act as axial
struts for the diaphragm bracing system which resists
lateral loads.
Joists run parallel to the slope and provide support
for roofing, deck, and superimposed loads. Bracing of
the joists, which are typically very unstable about their
weak axis, is provided by joist bridging designed and
supplied by the joist manufacturer. The bridging force
was estimated as 2% of the axial force in the joist chord
and was considered in design of primary truss top chords.
2.1.4 Diaphragm Bracing
Typically in structural steel and steel joist roof
construction, the metal deck is designed to act as a
diaphragm between framing elements. This diaphragm
28


action is used to provide bracing to resist lateral
buckling of top flange/chord of framing elements and to
resist applied lateral loads on the structure due to wind
and earthquake forces. As mentioned above, in the design
of the SLDCSF it was determined that commercially
available metal deck had inadequate capacity to resist
lateral forces with code-specified safety factors.
To resist the calculated forces, a system of
diagonal braces in the plane of the diaphragm was added.
Although this steel is light relative to other steel on
the job, each piece must be set with the crane. This
results in considerable crane time and a safety issue as
work must be performed at least 140'-0 above grade
without the metal deck installed. To address these
erection considerations, the bracing system shown in
Figure A-19 was selected. Bracing is fabricated in
multiple spans, 50' to 60' in length, and rests on top of
steel joists. Metal deck is interrupted at each line of
bracing and rests on the plate welded to the bottom of
the tube shape. Deck is then connected to the horizontal
plate to theoretically maintain diaphragm continuity.
Bracing was designed to remain stable in tension or
compression by increasing the 'y' axis radius of gyration
29


with the vertical plate between joists. The philosophy
of design is that since the diaphragm bracing is the
primary system upon which the design was based, it is
made continuous while the deck is interrupted.
2.1.5 Metal Decking
The metal deck selected for the structure is a 3
'N' type roof deck. Its gauge is controlled by moment
due to transverse superimposed gravity loads (roofing and
snow load). Although not designed for the lateral and
bracing loads, the deck has considerable diaphragm
capacity, particularly at ultimate (failure) conditions.
Metal deck is specified to be connected to supporting
structure using powder-actuated fasteners, more commonly
called "drive pins". These were selected based on ease
of work on a steeply sloped roof and since washers on
pins permit significant deformation of the deck without
losing uplift capacity. Despite interruption by
diaphragm bracing, fastening of metal deck at each
diagonal line was specified to maintain a working load
deck diaphragm capacity of 300 pounds per foot. This
30


level of shear is consistent with maximum forces
developed between lines of lateral bracing due to a code
level earthquake.
2.1.6 Perimeter Roof Bearings
The roof structure around the perimeter of building
bears on a series of slide bearings which permit it to
move freely under thermal expansion. An example of a
typical bearing is shown in Figures A-20 and A-21. The
slide bearings were provided since the concrete
supporting frame could not be designed to resist lateral
and thermal restraint forces within architectural and
economic constraints. Bearings were designed to
accommodate the maximum combinations of displacement
resulting from elastic level seismic loads of the
substructure and roof diaphragm (considered moving out of
phase), differential thermal change between roof and
substructure, shrinkage of concrete structure, and
cumulative construction tolerances. Bearings are to be
mounted to the roof structure at the working line so that
moment due to the displaced (eccentric) load would be
carried by the heavier cast-in-place system below.
31


Between the truss bearings on the straight sides of
the building, additional bearings are specified between
the perimeter ring beam and the concrete structure to
transfer lateral loads due to wind and earthquakes.
These bearings are centered along the length of roof
structure to minimize the amount of restraint due to
thermal displacements (i.e., the building can 'shrink'
and 'grow' about the center). An example of these
"guide" bearings is shown in Figure A-22. The bearing
was designed to resist loads parallel to the frame, but
apply a minimal amount of load (due to slide bearing
friction at interface) perpendicular to the frame.
2.1.7 Perimeter Ring Beam
The perimeter ring beam is a structural steel wide
flange shape at the base of the structure which supports
the first set of roof joists (refer to Figure A-23). The
ring beam is rigidly framed to the ends of the primary
truss at the bearings providing torsional resistance.
The ring beam slides in and out with the primary trusses
as the structure spreads under load. Joists which bear
on this member have bottom chords framed rigidly to the
bottom flange, limiting the amount of twist in the ring
32


beam between primary trusses. The ring beam acts as part
of the diaphragm bracing system and delivers the lateral
force due to wind or seismic loadings to the concrete
frame (refer to Figure A-22).
Due to geometry of the structure, the ring beam acts
as a tension ring around the structure. The structure
behaves very much like a braced dome, however, the
straight sides do not permit full "shell action to
develop. This behavior was studied using approximate
methods in the design office. In these analyses, a cycle
developed as follows: by analysis, the perimeter (ring)
beam has inadequate capacity to resist the axial tension
force associated with its stiffness. As the size is
increased, the tension ring becomes stiffer, thus
attracting more load. The net result is a structure
which deflects less, but is considerably more expensive.
This trend led to the introduction of expansion joints in
the ring beam to prevent this action.
2.2 Design Criteria
The roof structure of the SLDCSF was designed in
accordance with the Building Officials* Congress of
America (BOCA) model code, (BOCA, 1990), which is the
33


governing code for the City of St. Louis. Live loads due
to wind, snow, earthquake, and uniform floor loads
(catwalks) were derived from this document. Additional
requirements for rigging, scoreboards, speakers, etc.,
were determined from the Owner's requirements.
2.2.1 Roof Dead Loads
The SLDCSF roof system is a single ply, mechanically
fastened roof system over 3" of rigid insulation. This
system is fastened to a 3" type 'N' metal deck which is
supported by the joists, girders and trusses which are
defined in the Project Analysis Model. The overall
weight of the roofing system is about six pounds per
square foot (psf). In addition, the overall roof
structure is designed for a 2 psf allowance for catwalks
and a 5 psf allowance for the electrical grid, operable
convention grid and mechanical and plumbing services.
This results in a total superimposed uniform dead load of
13 psf, in addition to the self-weight of the structure.
34


2.2.2 Roof Gravity Live Loads
The 1990 BOCA code requires a uniform roof live load
of 15.4 psf to account for snow. In addition, unbalanced
snow loading due to wind-carried snow was considered,
with loads varying from 0 to 30.8 psf over the roof
surface. Refer to Figure 2-2 for profile of snow loading
used in design. Higher snow loads due to accumulated
snow in the "gutter" (refer to Figure 2-2) were applied
to the model. Individual catwalks were designed for 30
psf live load, but this load was not assumed to occur
simultaneously with maximum snow load. Finally, the roof
was designed for 150,000 pounds of rigging applied over
areas noted in Figures A-2 and A-3, and an 80,000 pound
scoreboard load both occurring simultaneously with
maximum snow loading.
2.2.3 Wind Loads
Wind loading for initial analysis model was
developed in accordance with BOCA using a 30-year mean
recourance and base wind speed of 70 m.p.h. An exposure
'B' classification was selected based on a downtown urban
35


uuc^u44^g£7 L&& pis&zivu
uu&kUfj Pf = PeIPg = (0.7) (1.1)20 = 15.4 PSF
Figure 2-2 Unbalanced Snow Load Profiles
36


setting. An importance factor of 1.07 (Category II per
BOCA Table 1112.3b) was used, resulting in the following
base pressure:
Pd = Pe I2Cp = 20.68 Cp (2-1)
Application of appropriate Cp terms resulted in the
wind forces defined in Figure 2-3. Actual wind forces
for final design are being refined using wind tunnel
testing, however, results were unavailable at the time of
writing this paper. Frequencies and mode shapes for the
roof structure were supplied to the wind consultant based
on the Design Office Model.
2.2.4 Earthquake Loads
For the initial analysis model, seismic loads were
calculated using the equivalent static force procedure
per BOCA 1990. The following parameters were used:
K = 1.0 Building system coefficient
(Shearwall system)
Z = .375 Acceleration coefficient (Zone 2)
S = 1.5 Soil/site coefficient
I = 1.25 Importance Factor
T = .504 sec. Period (from building calculations)
37








^Ry2APBT P*f
Design Wind Pressures per BOCA 1990
Pd = PcI2Cp = 20.68Cp
Figure 2-3 Summary of Wind Pressures
38


Applying these assumptions to the lateral force
equation, the resulting base shear coefficient is:
Vbase = (1.0) (.375) (.094) (1.25) W = 0.044 W (2-2)
where W represents the total weight of the structure.
Applying the upper triangular load distribution per
BOCA with no top force due to higher modes, the inelastic
equivalent lateral force at the roof in each direction
k
(nonconcurrently) is 910 For convenience of modeling,
this was applied as:
______Roof Shear_____ 910 kips 6.9% of weight
Total weight of roof 13,200 kips of all components
The final lateral analysis of the Design Office
Model is to include a dynamic (modal) analysis
considering interaction of separate substructure
buildings and roof.
2.2.5 Material Properties
Material properties specified for all members of the
structure are included in notes on Sheet A-l of Appendix
A.
39


2.2.6 Deflection Limitations
Roof structure is designed for overall maximum
deflection of (span length)/180 for total load
considering only bare frame stiffness. Deflection for
live load was limited to (span length)/360. Deflection
of individual components is limited to maintain 1/4" per
foot minimum roof slope under maximum load. Lateral
deflection of roof diaphragm between ends due to seismic
and wind loading is limited to (span length)/400
considering only stiffness of steel frame and diaphragm
bracing. The lateral deflection limit for roof diaphragm
did not consider racking deflection of columns from
highest seating level to roof bearing which is typically
limited by model codes (Uniform Building Code 1991) to
(height)/400. This is due to slide bearings at the
bearing points of the primary trusses which permit them
to move with respect to the cast-in-place structure below
with minimal friction (see Figure A-20).
2.3 Design Office Analysis
The following is a description of the Design Office
Model used to analyze the roof structure of the SLDCSF.
The author worked full time with two other professionals
40


in the preparation of the contract documents for this
structure. While there are no codified standards for
modeling and design of such structures, it is assumed
this work is representative of the "industry standard"
for such projects. The work was done by MARTIN/MARTIN of
Wheat Ridge, Colorado, a structural consulting
engineering firm with considerable experience in the
structural design of long-span structures. This
description is included to define areas of uncertainty at
the end of the design development phase which led to this
study.
The Design Office Model included only the primary
framing elements of the structure (see Figures 1-2 and 2-
4): trusses (primary gravity system) diaphragm bracing
(primary lateral system), and bottom chord bracing
(required for lateral stability). All members were
modeled as elastic beam elements with the exception of
cables for which slack conditions and prestress were
evaluated. Joists, joist girders, and metal deck were
not included in the model as their contribution to
overall behavior of the structure was initially assumed
to be minimal. The corner area, including radial joists,
41


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Figure 2-4 Plan View Key Map of Design Office Model
42


ring beam, and metal deck were not included due to
difficulty in modeling and the assumed participation
level.
2.3.1 Schematic Analysis Model
The structure was first modeled using a one-quarter
symmetric model with STAAD III (1989), a linear general
structural analysis program. This initial model included
only primary trusses and analyzed only gravity loads on
the structure. Due to symmetric boundary conditions,
unbalanced snow loading was not considered in the initial
analysis. STAAD III does not consider shear deformations
and no secondary (P-Delta) analysis is available for this
type of structure. The initial model was used for
comparing the various systems considered, and for initial
member size selection. A graphics plot of the STAAD III
model is included in Figure 2-5.
2.3.2 Design Office Model
After initial sizing, a full (all four quadrants)
model was developed. This Design Office Model included
all trusses, bracing, and the perimeter ring beam.
Corner joist beams and deck were not modeled. Figures 2-
43


at
Figure 2-5 Graphic Plot, Schematic Model


4 and 2-6 are geometry plots for the Design Office Model
showing a plan and perspective view of the model
respectively. The model was generated and analyzed using
LARSA (1991), a nonlinear (geometric only) general
analysis program. LARSA's elastic beam elements are 12
degrees of freedom elements, and the program
automatically assigns zero stiffness to any cables in
compression at each load step.
The Design Office Model derived member properties
from a built-in table of American Institute of Steel
Construction (AISC 1990) shapes. Built-up top chords and
diaphragm bracing were specified with prismatic
properties in the input file. The structure was analyzed
for self-weight, gravity dead and live loads, balanced
and unbalanced snow loads, and for lateral forces due to
wind and earthquake. Maximum load combinations were
derived from these forces and members designed.
All loads superimposed upon the primary truss
(weight of joists, deck, mechanical, snow, etc.) were
calculated by hand on a uniform load allowance basis and
applied as concentrated joist loads or distributed member
loads. Additional loadings and a response spectrum
analysis were completed later in design. For similar
45


Figure 2-6 Graphics Plot Design Office Model Perspective
View
46


(uniform) load cases, the Design Office Model
demonstrated excellent correlation with the initial STAAD
III model. Members were checked for stress in accordance
with the current AISC Allowable Stress Design
Specification (AISC 1990) using a program post processor.
2.3.3 Studies Involving Metal Deck Diaphragm
During design, the contribution of the metal deck
diaphragm was studied independently of Design Office
Model several times. In each case, an approximate
analysis was used to attempt to provide a bounded
solution. Typically, the lower bound was represented by
the Design Office Model (i.e., no stiffness associated
with deck). The upper bound was evaluated as the
limiting strength of the metal deck diaphragm. The study
of the PAM and successive models was undertaken to refine
these approximate analyses.
The metal deck depth and gauge were selected to be
adequate to support gravity loads and wind uplift
pressure normal to its surface. In addition, the deck
fastening was selected to provide adequate strength to
resist in-plane forces due to wind or earthquakes between
lines of structural diaphragm bracing. Strength and
47


stiffness calculations for the metal deck diaphragm were
made in accordance with provisions of the SDI Diaphragm
Design Manual (Luttrell, 1987).
In addition to the diaphragm calculations of SDI,
the flexibility of the deck to primary support connection
was evaluated. As noted previously, researchers on metal
deck diaphragms found this to be a major contributor to
diaphragm flexibility. These calculations are included
in Appendix B. Due to the depth reguired for consistent
detailing, these calculations indicated that joist seats
were inadequate to transfer shear from deck to the
primary trusses. A transfer plate/continuous deck
support arrangement (see Figure A-24) was added. This
system is approximately two orders of magnitude stiffer
than metal deck, hence it was not combined with other
factors in computing diaphragm flexibility.
2.3.3.1 Evaluation of Thermal Restraint
The first area studied was thermal stresses which
might develop in the diaphragm due to restraint by the
primary trusses. For this condition, metal deck which is
exposed to sunlight during the construction period heats
up quickly due to exposure and low thermal mass. The
48


supporting structure below (in this case the primary
trusses) is shaded by the deck and, hence, has little
solar gain. Although this is a temporary problem during
construction, it is important to evaluate. The author
has observed this behavior on other projects and in
extreme cases it may fail deck-to-support connections.
In evaluating this condition, axial stiffness of the
deck is of primary importance as opposed to shear
strength. Perpendicular to its corrugations, the metal
deck can flex, thus limiting restraint and therefore
stress (see Figure 2-7). Parallel to the corrugations,
the deck acts as a column between supports. With no
restraint, the deck expands freely; however, at points of
connections to primary trusses, very high shears may
result. Calculations in Appendix B represent the
approximate evaluation of the thermal
restraint/differential temperature problem.
2.3.3.2 Tension Ring/Shell Action
The second area studied was the participation of the
corner joists, deck and ring beam acting as a
shell/tension ring structure. It was recognized early in
design that as the primary trusses deflected vertically,
49


Perpendicular to Corrugations
Dfcti F&F\[E rY&Z-To etfov,\Xf&
to feUNU^HT1eHowM eoup,
D^&HE? uiue f?EPEE6&45> EvC&aEB&TB?
DEFLECTED £+4A|=ep KELlEVfe THE^M^L
S£ETVJESJ
Parallel to Corrugations
_ MerTuLce^K.-
2
nse
/
Jtfleps
Joisr&raP^IN^
V INPioqs£> tso^-io -Jo^r
<2otvjis|B2Tk7N POINTS.
ToPonopp g#|'MB^UP6b(i,
MET^L DEck feHEST leitfjlPLY
f=^HNEP to JoieTs AT EMM
£>UffSPfEr. DE^I^&STIWEP
F=|eoM E*pAL|S|OW BY NPf&RTlNLSf
Figure 2-7 Response of Metal Deck Subject to Differential
Thermal Changes
50


the bearing points (bottom chord bearing trusses) would
move out. This tendency indicated that the ring beam at
each corner (constructed in seven segments passing
through 90) would either have to lengthen as the roof
spread or its geometry must change to accommodate these
displacements.
The approximate analysis of the ring beam at the
corners is included in Appendix B. In this analysis, it
was assumed that the truss deflections were unaltered by
the presence of the continuous corner structure and the
diaphragm. The shape of the structure was assumed
unchanged with only the individual pieces of the
segmented corner lengthening as required to match the
displacements of the primary trusses. This assumption
implies that the deck and joists of the diaphragm are
very rigid compared to the axial stiffness of the ring
beam and force it out uniformly. This evaluation
indicated that strain in the collector was far beyond
acceptable levels, hence requiring a joint in the
collector beam.
Several different schemes were evaluated for
jointing. The scheme which was selected was developed
based on an approximate analysis of strain variation at
51


the corner diaphragm. The joints were extended from the
roof perimeter towards the center of the structure to a
point where shear stress in diaphragm did not appear
critical. Two expansion joints in the diaphragm at each
corner were detailed. This joint served to relieve axial
stresses in the ring beam (see Figures A-25 and A-26),
while relieving shear stress in the diaphragm.
The evaluation of stresses in the diaphragm were
made based on calculating relative displacements of the
corners of the individual pieces after jointing.
Displacements of trusses were taken directly from the
Design Office Model analysis, while displacements of
joists were estimated based on depth requirements and
loading criteria on the drawings. This method of
analysis did not account for increased flexibility of the
diaphragm due to its initial warped shape. As a result
of this approximate analysis eight 113'-4 joists and
eight expansion joints were added in the deck,
insulation, and roofing. Refer to Figure A-5 for
location of joints at corners.
52


2.3.3.3 Deck Warping
The next issue studied was warping of the'roof deck
at the north and south ends of the SLDCSF. This portion
of the roof structure is supported by three segment
sloped joists spanning 193'-9" (see Figure A-7). The
warping of the deck results from the unequal vertical
displacement along the length of the primary truss at the
upper bearing of the joists. Since the lower end is
restrained vertically, but free (with the exception of
friction in the slide bearings) to move horizontally, the
lower ends of the joists "kick out" a differential amount
along the length. This phenomenon is discussed more
thoroughly in Section 3.2.3. The approximate analysis of
this condition is included in Appendix B. These
calculations indicate a deck shear level exceeding the
allowable at the first two joist spaces adjacent to the
primary north-south trusses. To limit stresses at these
locations, two deck relief joints (see Figure A-27) at
each end were added. Refer to Figure A-5 for plan
location of relief joints. The east and west sides of
the structure were also evaluated for this condition.
The level of shear in the deck was predicted to be
53


acceptable since the relative displacement of north-south
truss top chord is small with respect to the primary
trusses panel points in the east-west direction.
2.3.3.4 Approximation of Deck Lateral Stiffness
Finally, the lateral stiffness of the metal deck
diaphragm of the SLDCSF was considered. The lower bound
(no stiffness) was modeled by the Design Office Model.
The upper bound stiffness was calculated in accordance
wiht SDI based on the stiffness of a planar segment of
diaphragm 84'-4" x 64'-7, with the fastener pattern for
the deck as noted on the drawings. The axial stiffness
of the diaphragm bracing in the Design Office Model was
then increased to simulate the deck stiffness. The model
was rerun with new displacements and forces calculated.
This method of evaluating lateral stiffness for the
diaphragm did not account for added stiffness due to
continuity at the corners.
2.3.3.5 Evaluation of Drag Strut
In the Design Office Model, the perimeter ring beam
is assumed to receive load only from the diaphragm
bracing. With metal deck continuously attached to this
54


beam via a transfer plate (refer to Figure A-23), some
lateral load carried in the deck will be delivered to
this beam creating a "drag strut".
Design of this drag strut was done using a bounded
solution. In the computer analysis the member forces
were developed assuming all lateral load is delivered to
the ring beam via the diaphragm bracing and transferred
to the bearings as per the Design Office Model. In the
upper bound analysis, force distribution was considered
assuming metal deck carries the entire shear along the
full length of the structure in a uniform manner,
resulting in higher drag (axial) forces away from the
bearings. The bounded analysis resulted in a relatively
uniform sized member for the ring beam around the
structure.
2.4 Project Analysis Model
A more detailed model of the SLDCSF roof was
constructed to attempt to address the unknowns of the
Design Office Model regarding the diaphragm behavior.
This model includes all of the elements of the Design
Office Model and the joists, joist girders, corner
structure, and metal deck diaphragm. A one-quarter model
55


with symmetric boundary conditions was selected to limit
calculation time and redundant output. The selection of
this model limited evaluation of the structure to uniform
load cases. This was deemed acceptable due to the
uncertainties associated with the materials modeled, the
general nature of evaluation of results, and the answer
sought. As documented by Luttrell (1981a), the variation
in strength and stiffness of a series of "identically
constructed" diaphragms is large. This results in the
rather high safety factor of 2.5 recommended in the
design of metal deck diaphragms by SDI. In light of
these facts, it must be established that this model was
constructed to look at gross behavior of the SLDCSF roof
structure. The goal is to identify or verify trends in
the model, not to establish a precise stress level at a
point in the diaphragm.
2.4.1 Description of Project Analysis Model
The Project Analysis Model was constructed using
SAP80 (1984), a linear general analysis program for
static and dynamic analysis of structures. SAP80
features a 12 degree of freedom elastic beam element and
evaluates secondary effects (P-Delta) using a negative
56


geometric stiffness matrix applicable to any geometry.
The program also supports four node membrane type finite
elements, which were used to model metal deck stiffness.
A complete listing of the SAP input file, along with
output files referenced in this paper, are included in
Appendix D. A key map, describing geometry, joint
numbering, member numbering, and element numbering, is
provided in Figure 2-8.
2.4.1.1 Model Geometry
The corner areas of the SLDCSF are modeled with the
actual warped geometry of the structure. As can be seen
in Figure 2-8, joint 286 and joint 538 create the end
points of a constant elevation line across the corner
area. Since the joists running down the slope have a
constant lower bearing elevation, and a variable length,
this results in warping of the diaphragm in each joist
space. The area bounded by joists 13 and 117 on the
south, and by 370 and 532 on the north, also represent a
warped deck surface bearing on the upper segment of the
triple pitched long span joists.
57


2.4.1.2 Member Properties
Truss members, ring beam elements, and diaphragm
bracing were all coded with identical section properties
to the Design Office Model. All members were described
as prismatic members since SAP80 does not support the
data base of AISC shapes. Steel joist properties were
input as eight degree of freedom elastic beam elements,
omitting shear areas around the two principal axes.
Section properties are not typically published by joist
manufacturers, rather, joists are selected from Standard
Load Tables (Steel Joist Institute, 1992) published by
the Steel Joist Institute (SJI). This results in
specifying a minimum stiffness and strength value and
giving the joist manufacturer maximum flexibility in
constructing the joists. Based on these load tables,
approximate areas and moment of inertia were calculated
for each joist type. The section property calculations
are even more approximate, when considering the large
number of "special" joist types specified on the
drawings. These special joists (see Figure A-7) have
multi-pitched geometry and variable depth with only a
minimum centerline depth reguired of the fabricator. The
multi-pitched joists were assumed to be constant
58


stiffness, kinked elastic beams for the Project Analysis
Model. Area and strong axis moment of inertia were
calculated based on specified centerline depth. Based on
the area calculated, the approximate chord angle size was
determined. The weak axis stiffness of the joists was
calculated based on the chord angle sizes determined
above, assuming only equal leg double angles with a 1"
gap will be used to construct them (typical of steel
joists). In some locations (joist types 34 and 35 for
example), weak axis section properties were exaggerated
to maintain stability. Joists in the real structure are
typically braced by rows of bridging at spacings defined
by SJI to maintain stability. Bridging is in turn
anchored to the adjacent structure (in this case the
primary trusses) or x-braced at end bays to maintain
stability. These exaggerated section properties are
representative of actual weak axis stiffness of joist
chords with bridging installed.
Joists at the north end of the structure in the
Project Analysis Model match the actual spacing specified
on the construction documents (see Figures A-5 and 2-8).
On the east side, and in the corners; however, joists
coded in the Project Analysis Model represent "equivalent
59


joists" that include the stiffness of two or more joists
in the actual structure. These equivalent joists were
developed to maintain a reasonably uniform finite element
mesh in modeling of the roof diaphragm. Parameters
defining the fineness of the element mesh are described
in Section 2.4.1.5. Calculations describing the
development of section properties of equivalent joints
are included in Appendix C. On the east side, five
equivalent joists were used in each 84'-4" bay in lieu of
the six joists in the actual structure. This arrangement
permitted the intersection of the finite element mesh
lines at equal elevation, and the diaphragm bracing to
coincide.
As a convenience of modeling, at the corner areas,
all joists were coded to have either the lower bearing
frame into a vertex point of the perimeter ring beam, or
the upper bearing frame into a node point in the element
mesh. In six locations (see Figure 2-8) this resulted in
a joist which is bent in plan at the last panel to
maintain the mesh. At these locations, additional struts
(coded as 900 series members in the Project Analysis
Model) were added to maintain the lateral position of the
joist. Finally, joists were typically released at
60


bearing points, thus correctly modeling their simple span
behavior. Joists which bear on the perimeter ring beam
were not released to be consistent with the joist-to-beam
connection detailed (see Figure A-23). This condition
maintains stability of the ring beam under torsional
loads.
2.4.1.3 Metal Deck Diaphragm
The metal deck diaphragm of the SLDCSF was modeled
as membrane finite elements in the Project Analysis
Model. In typical building applications, metal deck is
sized for bending due to gravity loads and then evaluated
for diaphragm strength due to lateral loads. In this
process, the participation of the metal deck diaphragm in
resisting gravity loads is neglected. This is acceptable
for a typical flat roof or single plane diaphragm,
however may not be appropriate for a roof such as the
SLDCSF. In this structure, displacement of the roof
structure due to gravity loads induces shear within the
diaphragm.
It would be difficult if not impossible to
accurately describe the stress level in the diaphragm due
to gravity loads. With conventional erection methods, no
61


decking is present on the structure until displacement
associated with self-weight of the primary trusses,
joists, and some of the deck has occurred. Until the
entire diaphragm is installed and fastened, the
distribution of stress in the diaphragm will vary. In
addition, the presence of any temporary shoring will
further change the displacement pattern, therefore the
stresses induced. For this project, only stresses
arising from loading after complete erection of the deck
are considered. The superimposed dead load case in the
Project Analysis Model assumes that the entire structure
(trusses, joists, etc.) is completed and any shoring used
in erection has been released. Furthermore, it is
assumed all deck is in place and completely fastened.
The weight of superimposed dead load (insulation,
roofing, and mechanical and electrical services) are
applied to the roof as a uniform load of ten pounds per
square foot (psf). The forces due to the uniform snow
load of 15.4 psf are determined by scaling the dead load
case up by (15.4+10)/13 = 1.94. In the Project Analysis
Model, uniform load was applied as a body force
multiplier of the membrane elements.
62


2.4.1.4 Development of Membrane Element
Properties
Due to the corrugated configuration of metal deck
sheets, a metal deck diaphragm is significantly more
flexible than a steel sheet of equivalent thickness. The
cantilevered diaphragm shown in Figure 2-9 can be used to
demonstrate this principle. Properties of metal deck and
attachments to supports shown in Figure 2-9 are identical
to the metal deck diaphragm system specified for the
SLDCSF roof. The notation is consistent with the SDI
Diaphragm Design Manual (Luttrell, 1987).
The average shear stress in the panel is T = P/Lt,
while the average shear strain is K = A/a. Based on
these equations, for a uniform thin steel sheet, the
shear modulus could be expressed as:
G =r/y = P/Lt (a/A) (2-3)
In testing of metal deck diaphragms, 'L', 't', and
'a' are a function of the geometry of the diaphragm while
' P' and 'A' can be read from the load cell and dial gage
(or similar apparatus) directly. Results of testing such
diaphragms indicate that G is typically in the range of
63


Deck Stiffness
For 20ga Deck G1
105.6
4.31 + 0.3(488)/12 + 3(12){.251)
= 41.3 k/in
For 22ga Deck G =
870
= 28.5 k/in
4.31 + 0.33(653)/12 + 3(12)(.288)
For 21ga Deck G1 3 5.0 k/in
Figure 2-9 Typical Cantilevered Diaphragm
64


900 ksi to 1100 ksi. From mechanics of materials G of an
isotropic element is known to be:
G = E (2-4)
2 (1+1^)
which is approximately 11,200 ksi for steel.
To account for this discrepancy, SDI quantifies the
diaphragm shear stiffness as:
G' = T Geff = (P/a) (a/L) (2-5)
where GEFF is the empirical shear modulus determined from
testing. Since it is expensive and undesirable to
require full scale testing of all profiles and fastener
configurations of a metal deck to determine reliable
diaphragm values, SDI has developed equations for
analytical determination of G'. The stiffness of any
diaphragm can be evaluated using the equation:
G' = _________Et________ (2-6)
2 (1+j^) s/d +Dn +C
Plugging in empirical data from charts in the SDI
Diaphragm Design Manual (Luttrell, 1987) into equation 2-
6, the stiffness of the decking on the SLDCSF is 35.0
kips/in. Refer to Appendix C for complete calculations.
Comparing this value with the actual shear modulus from
equation 2-4, it is seen the metal deck diaphragm is
65


about l/10th as stiff as a flat sheet of steel of the
same thickness.
G1 :D1 = ____35.0 k/in_________ = 0.0933
G' (11,4000 k/in ) (.0329 in) (2-7)
To account for this fact, the membrane element thickness
was reduced by this ratio. Therefore, the membrane
thickness specified in the Project Analysis Model is:
T = (.0329 in)(.0933) = .0031 in (2-8)
where .0329 represents the thickness of the steel sheet
(21 gauge). For this project, the formulas and
recommendations of SDI were used in developing the model.
Refer to the Diaphragm Design Manual (Luttrell, 1987) for
a complete treatment on this subject.
Since the analysis was completed on a linear
analysis program, it is implicitly assumed that all
membrane elements perform linearly regardless of stress
level. Figure 2-10 shows a typical load-deformation
curve from a full scale diaphragm test (Lutrell, 1981a).
The behavior is nearly linear from no load up to
approximately 50% of the ultimate load. The Steel Deck
Institute calculates stiffness at 40% of the ultimate
load, consistent with its 2.5 safety factor. Beyond this
level, the curve flattens at a rate dependent on deck
profile and diaphragm geometry. It can be seen
66


APPLIED LOAD (KIPS)
Figure 2-10 Typical Load/Deformation Curve for Metal Deck
Diaphragm
67


from the shape of the curve, that as the shear in the
diaphragm moves beyond the linear range, the diaphragm
becomes more flexible. This indicates that any locations
with very high shears predicted by the Project Analysis
Model are probably overestimated.
A final note on the membrane element in the Project
Analysis Model. Considered alone, metal deck is an
orthotopic material, being significantly stiffer parallel
to the corrugations than perpendicular to them. Research
done at Cornell (Nilson 1960) specifically studied if
full scale diaphragm panels with varying aspect ratios
had different stiffness parallel and perpendicular to the
corrugations. This research indicated the diaphragm
assemblies behaved isotopically despite the orthotopic
deck. The current SDI design recommendations (Luttrell,
1987) make no differentiation between stiffness parallel
and perpendicular to the deck. This topic was further
discussed with Dr. Lutrell (Lutrell, 1992) in preparation
of this paper, and he indicated micro finite element
modeling of corrugated sheets by Davies (1977) and others
had been inconclusive. Based on these recommendations,
an isotropic membrane element was used.
68


To further verify the membrane element, three small
computer models were built. Each model was coded to
match the geometry and stiffness of a full scale planar
diaphragm test done by the Steel Deck Institute (Luttrell
and Huang, 1981). In this way, correlation between load
test, SDI formulas, and the author's finite element model
could be compared. Correlation of computer model
deflection/SDI deflection was 1.254/ 1.31, and 1.35, for
the three tests respectively. Correlation of computer
model/load test was 1.066, 0.812, and 1.09. Although the
variation of 35% seems large, it is consistent with the
correlation found between calculation and load test found
by SDI in the first edition of the Diaphragm Design
Manual (Luttrell, 1981). This variation is important as
it limits the refinement of conclusions of the study.
More than any other factor, uncertainties in modeling the
metal deck diaphragm limit evaluations to gross behavior
of the diaphragm or trends at best.
2.4.1.5 Development of Finite Element Mesh
The initial finite element mesh selected for the
analysis is shown in Figure 2-11. The mesh was selected
based on convenience of modeling. Each of the "bays"
69


which bound the diagonal diaphragm bracing was divided
into a six by six grid. This arrangement allows the
corners of the individual element to coincide with
diagonal bracing. Elements at the corners were oriented
to maintain equal elevation contours.
After review of the membrane stress output for the
first run, it was determined that the mesh was
inadequately refined at the east side/corner interface.
This was based on large variations of shear at a joint
reported by adjacent elements. The mesh was refined to
that shown in Figure 2-8 yielding improved results.
Although a few joints with poor stress correlation exist
in the final model, in general the results are adequate
for this study.
2.4.2 Additional Analysis Models
The Project Analysis Model of the SLDCSF is a model
with no deck joints and a continuous tension ring. All
documentation up to this point has been in support of
this model. The boundary conditions of this one-quarter
symmetric model are appropriate for gravity loading only.
After evaluation of the output from the Project Analysis
Model, additional models were required to check the deck
70


jointing of the Design Office solution, and to provide a
model with appropriate boundy conditions to analyze
lateral forces. Table 2-1 defines each of the analyses
made and how it differs from the Project Analysis Model.
In each case, a figure reference for the key map is
included. Results of these models are discussed in
detail in the next section.
71


TABLE 2-1 Summary of Analysis Models
FILE NAME FIGURE # FOR KEY MAP VARIATION FROM PROJECT ANALYSIS MODEL REMARKS
SL5 2-8 None This is the Project Analysis Model Baseline
SL6 2-12 Added relief joints adjacent to corners. Support angles at relief joints have bending stiffness in plane of diaphragm. Used to determine if relief joints alone are adequate to reduce deck shears to allowable levels.
SL7,SL7A 2-8 (refer to Section 3.3.3 for revised boundry conditions) Revised boundary conditions for lateral load. Added Y direction body force multiplier. Added spring supports to accurately model drag forces at ring beam. Used for analysis of lateral loads. File SL7A includes deck jointing shown in Figure 2-13 for SL8
SL8 2-12 Modified SL6 to incorporate joint at tension ring. Used to determine if design office joints are adequate to alleviate deck overstress.
SL BARE 2-8 Baseline run without shell elements. Used to compare deflections due to structure selfweight with design office model.
SL BARE2 2-8 File SLBARE with joint in tension ring. Used to compare deflections due to structure selfweight with design office model.


RPR JoiKp3>Ksm_ PELSiee
AT-mie
r?iur im
LEGEND
xx, xx
Figure 2-12 Partial Key Map for Project Model with Relief Joints
INDICATES JOINT NUMBER. FOR OVERALL FIGURE
CLARITY JOINT NUMBERS ARE NOTED AT THE
BEGINNING AND END OF ROWS OF SEQUENTIALLY
NUMBERED JOINTS.
INDICATES ELASTIC BEAM ELEMENT MEMBER
NUMBER.
INDICATES MEMBRANE ELEMENT NUMBER. THE
MEMBRANE IS PRESENT OVER THE ENTIRE
SURFACE. FOR OVERALL FIGURE CLARITY
ELEMENT NUMBERS ARE NOTED A FIRST AND LAST
ELEMENT OF A SEQUENTIALLY NUMBERED ROW OF
ELEMENTS.
INDICATES BOUNDRY CONDITIONS ENFORCED ON
THE JOINT NOTEO. FX, FY, FZ DENOTE A
RESTRAINED CONDITION IN THE GLOBAL X, Y OR
Z DIRECTION RESPECTIVELY. MX, MY, MZ
DENOTE A RESTRAINED CONDITION IN THE GLOBAL
X, Y OR Z DIRECTION. DEGREES OF FREEDOM
NOT NOTED ARE UNRESTRAINED.
j
AREA'S WiT-HSUT AND
MBMBEEta STEP APE
identical jo EICjUPE 2-8
Rp* joiwrnv Mem-
reusAee a*ial,m*
AWP ME ATT14L& Rptkff
IWT^WeiDtJ PJIJCf


3. Evaluation of Analysis Results
All of the analysis models described in the previous
section were evaluated in detail and compared with the
Design Office Model (DOM). For gravity load analysis,
the Project Analysis Model (PAM) was stiffer than the
DOM. As jointing was added, deflection increased to
correlate more closely with the DOM. The pattern of
shear force in the diaphragm was largely unchanged by the
introduction of joints, however, the magnitude of shears
in the local region of the joints increased
significantly.
The load on DOM was somewhat more than on the PAM.
A summary of reactions at primary truss bearings is
provided in Table 3-1 for each model. The ratio of
PAM/DOM varies between 0.85 to 0.94 at the various
bearings. The differences between the two models are
primarily due to two sources as follows:
1) Joist self-weight loads for the DOM were
calculated on a tributary sguare footage times
a unit load basis. For the short span joists
on the east side, an allowance of five psf was
used. The actual self-weight on the PAM is
74


TABLE 3-1 Summary of Gravity Load Reactions at Bearings
JOINT NUMBER SUPERIMPOSED DL ON PAM (KIPS) SELF WEIGHT ON BARE FRAME (KIPS) PAM TOTAL (KIPS) DESIGN OFFICE MODEL (KIPS) PAM/DOM
352 356 495 851 876 0.94
358 344 446 790 897 0.88
364 295 419 714 844 0.85
544 201 290 491 559 0.87
Total Truss Reactions 2846. 3176 0.896


30 PLF or 2.14 psf. Variation was more extreme
at the corners.
2) Superimposed unit load of deck, joists and
roofing in the area tributary to primary truss
top chords was coded twice in the DOM. This
represents a 14% error in superimposed dead
load to the typical east-west primary truss top
chord.
It was decided that an accurate distribution of
self-weight in the PAM was more important than matching
the DOM precisely. Comparisons of the forces between the
two models are primarily to validate the PAM. To
compensate for the loading differences, deflections in
Tables 3-1 through 3-4 for the DOM have been modified
representative of PAM/DOM reaction at that truss. For
evaluation of forces and stress, a composite factor of
0.89, consistent with the total load ratio per Table 3-1,
was used.
The lateral analysis model was analyzed with and
without the deck jointing in place. The analysis without
deck jointing yielded only 50% of the maximum lateral
displacement, following a similar pattern to the DOM.
The diaphragm and bracing worked together in carrying the
76


lateral force to the support. No conditions of metal
deck diaphragm overstress were predicted by the model.
The analysis with deck jointing in place had lateral
deflections similar to the DOM in pattern and magnitude.
Several locations of diaphragm overstress were predicted
by the model.
Use of the one-quarter symmetric model for the PAM
required different boundary conditions for lateral and
gravity loading. For each boundary condition, the
unjointed model is reviewed in detail and compared with
the DOM. The variations from these models with the
introduction of joints is then discussed.
3.1 Evaluation Criteria
The DOM was built and analyzed by three individuals
working independently, using three different analysis
programs. The three analyses included lateral and
gravity runs, nonlinear behavior, and dynamic analysis as
described in Section 2.4. Since the same results for
member stress and deflections were achieved, these values
were used as a baseline to check the PAM.
Strength of the metal deck diaphragm was evaluated
in accordance with SDI criteria. SDI diaphragm formulas
77


are written for the typical case of planar diaphragms
subject to wind or earthquake loading (i.e., loads which
are transient or temporary in nature). For permanent
loads causing deck shear (i.e., due to gravity or
lateral earth preassure), SDI recommends reducing the
allowable loads by 25%.
The PAM was analyzed using a superimposed dead load
of 13 pounds per square foot (psf). The actual load
after all decking is complete (refer to Section 2.4.1.3)
is ten psf. Evaluation of shears for the dead load
condition were made based on a ratio of 10/13 = 0.76
since the PAM is linear. The SLDCSF was designed for a
15.4 psf uniform snow load, hence evaluation of diaphragm
shear forces was made based on a ratio of 15.4/13 = 1.18
times the output forces. For total load, the shears are
evaluated for 1.94 times the output forces.
Based on these criteria, the following values where
used to define "overstress" for allowable and ultimate
load capacities:
Working Stress Ultimate
SDI
Gravity Load 225 plf 562 plf
Lateral Load 300 plf 750 plf
78


Ultimate loads were calculated as 2.5 times the SDI
working stress allowable loads (see Figure 2-10).
In evaluation of diaphragm shears, it is important
to review two points discussed earlier. First, an
indication of shear values in excess of SDI allowables
does not indicate failure of the deck from a
serviceability standpoint. The ultimate capacity of the
deck is typically two and one-half times the SDI
allowable loads (refer to Figure 2-10) As the allowable
load is surpassed, the deck softens relative to the steel
frame, hence, the PAM probably overestimates the shear
force. Second, the strength of metal deck diaphragms is
inconsistent due to the large variation in the quality of
individual connections. SDI analysis methods compare the
average shear across the depth of a rectangular diaphragm
to the specified allowables. The direction of shear and
extent of the overall diaphragm used in calculating this
average value influences the acceptability of the
diaphragm. SDI gives no guidance for a complex folded
roof such as the SLDCSF. In this study, shear contours
are shown in figures without averaging and all values are
rotated to a common global axis. "Overstress" areas in
Figures 3-5, 3-14, and 3-15 are defined as locations
79


where average shear over the planar segments of diaphragm
parallel to the joists span exceed SDI allowables.
Absolute magnitudes of shear at any point in the
diaphragm have little meaning. Rather this study is an
attempt to evaluate general trends and patterns.
3.2 Gravity Load Project Analysis Model
The PAM has no joints in the metal deck diaphragm or
the tension ring. It represents the behavior of the roof
prior to introduction of joints. The comparison of the
results of this model with the DOM demonstrate the
participation of the metal deck diaphragm in stiffening
the overall roof structure. Figure 3-1 is a quarter plan
of the SLDCSF summarizing the analysis information for
the PAM. The mesh shown is identical to the model key
map shown in Figure 2-8. The magnitude of shear in the
diaphragm is shown via contour lines with forces in
pounds per linear foot (plf). All element shear stress
was transformed to the axis noted in Figure 3-1,
consistent with SDI analysis. The axial forces in kips
carried by bracing members and joists referenced by this
paper are noted adjacent to the member.
80


3.2.1 Deflection
The PAM predicts significantly less deflection for
the same loads than the DOM. Table 3-1 shows horizontal
displacements at slide bearings due to gravity loads for
the DOM, the PAM, and the Project Model with jointing
introduced. Deflection listed represents displacements
under full self-weight and superimposed dead load (no
live load). Displacements for the two project models
were derived from the addition of two terms as follows:
1) Displacement of bare frame model with no
diaphragm under self-weight of frame.
2) Displacement of PAM with diaphragm in place
under full superimposed loading.
In this manner, the deflection due to self-weight of
the structure prior to installation of decking is not
evaluated based on the stiffer model which includes the
diaphragm.
Comparison of the displacement shows that, in
general, the PAM predicts approximately 10% less
deflection than the DOM. The PAM predicted Joint 490 to
have more displacement than the DOM. This value is
81


TABLE 3-2 Lateral Displacement at Slide Bearings Due to Dead Loads
All values shown in inches
JOINT NUMBER 352 358 364 382 436 490 544
Direction of Primary Displacement Y Y Y X X X X
Superimposed Dead Load on PAM 0.888 0.913 0.752 1.395 1.397 1.286 0.789
Self-Weight on Bare Frame 1.425 1.305 1.092 2.063 2.031 1.850 1.117
Total PAM 2.313 2.22 1.84 3.46 3.428 3.136 1.915
Design Office Model (DOM) 2.61 2.67 2.44 4.48 4.24 3.38 2.41
Factor 0.94 0.88 0.85 0.87 0.87 0.87 0.87
Modified DOM 2.45 2.34 2.07 3.90 3.69 2.94 2.160
PAM/Modified DOM 0.94 0.94 0.89 0.89 0.92 1.05 0.89
Superimposed Dead Load on Jointed Model 0.888 0.929 0.818 1.457 1.475 1.376 0.921
Self-Weight on Bare Frame Jointed Model 1.425 1.33 1.21 2.182 2.176 2.001 1.294
Total Jointed Model 2.313 2.26 2.028 4.486 4.244 3.380 2.481
Jointed Model/ Modified DOM 0.94 0.97 0.98 0.93 0.98 1.14 1.01


influenced by the relatively heavy self-weight of the
joists at the north end when compared with the unit load
allowance in the DOM.
Table 3-2 also illustrates a change in the pattern
of displacement predicted. The presence of the tension
ring/shell system restrains the trusses on the east side
increasingly moving from Joint 352 to Joint 364. This
change in deflected shape pattern is shown schematically
in Figure 3-2.
Table 3-3 summarizes the vertical displacements of
primary truss joints for each of the three models.
Displacements for the project models were derived using
the two-step method described for Table 3-2 above. The
PAM predicts the structure will deflect between 84% and
109% of the value analyzed in the modified DOM. The
relative deflections are summarized on a plot in Figure
3-3, showing a trend for the ratio of the PAM/DOM
deflections to become greater towards the center of the
span. This may be a function of increasing relative
stiffness of the metal deck diaphragm (as the slope of
the roof becomes steeper) vs the tied arch truss/bracing
83


TABLE 3-3 Vertical Displacement at Primary Truss Joints Due to Dead Loads
All values shown in inches
JOINT NUMBER 40 46 52 118 124 130 196 202
Superimposed Dead Load on PAM 4.79 4.77 4.19 4.41 4.33 3.36 3.51 3.84
Self-Weight on Bare Frame 7.89 6.91 6.53 7.44 7.14 5.39 5.74 5.44
Total PAM 12.86 11.68 10.72 11.85 11.47 8.75 9.25 9.28
Design Office Model 14.56 15.36 14.77 14.33 14.40 11.90 11.43 11.27
Factor 0.94 0.88 0.85 0.94 0.88 0.85 0.94 0.88
Modified DOM 13.68 13.52 12.55 13.47 12.67 10.12 10.74 9.92
PAM/Modified DOM 0.94 0.86 0.85 0.87 0.91 0.86 0.86 0.94
Superimposed Dead Load on Jointed Model 7.89 7.05 6.96 7.44 7.29 5.86 5.73 5.55
Self-Weight on Bare Frame Jointed Model 4.79 4.84 4.43 4.41 4.41 3.62 3.49 3.90
Total Jointed Model 12.68 11.89 11.39 11.84 11.70 9.48 9.22 9.45
Jointed Model/ Modified DOM 0.93 0.88 0.91 0.88 0.92 0.94 0.86 0.95


TABLE 3-3 Continued
JOINT 208 274 280 286 424 430 532 538
Superimposed Dead Load on PAM 3.17 2.04 2.65 2.01 5.52 3.97 3.81 2.49
Self-Weight on Bare Frame 4.38 3.27 3.32 2.72 6.88 4.55 4.87 3.26
Total PAM 7.56 5.31 5.97 4.73 12.40 8.52 8.68 5.75
Design Office Model 9.78 6.75 6.52 5.50 14.05 9.0 11.0 6.53
Factor 0.85 0.94 0.88 0.85 0.87 0.87 0.87 0.87
Modified DOM 8.31 6.34 5.74 4.67 12.22 7.83 9.57 5.68
PAM/Modified DOM 0.91 0.84 1.03 1.01 1.01 1.09 0.91 1.01
Superimposed Dead Load on Jointed Model 4.69 3.27 3.35 2.86 7.17 4.69 5.41 3.59
Self-Weight on Bare Frame Jointed Model 3.36 2.30 2.67 2.09 5.67 4.05 4.11 2.74
Total Jointed Model 8.05 5.57 6.02 4.95 12.84 8.74 9.52 6.33
Jointed Model/ Modified DOM 0.97 0.88 1.05 1.06 1.05 1.11 0.99 1.11


Figure 3-2 Horizontal Deflected Position at Slide Bearings Due to Gravity Loads



03
I

Figure 3-3 Vertical Displacements of Primary Trusses Due to Gravity Loads


system. By stiffening the ends, less displacement occurs
at the center. Additional studies would be needed to
verify this conclusion.
The vertical deflections of the project model with
joints follow the same general pattern as the PAM. For
this more flexible configuration, ratios of displacement
to the DOM vary between 86% and 111%.
3.2.2 Member Forces
Member forces were compared between the DOM and PAM.
In general, axial forces in the primary trusses predicted
by the PAM were between 6% and 15% lighter than those of
the adjusted DOM. This is consistent with the reduced
deflection of the primary trusses predicted.
The forces in the diaphragm bracing predicted by the
PAM vary between points of connection to the primary
truss (refer to Figure 3-1). This behavior was not
observed in the DOM since joists and deck were not
modeled. A typical joint demonstrating the
joist/deck/bracing interaction is shown in Figure 3-4.
The components of the change in force in the diaphragm
88



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Figure 3-4 Metal Deck/Diaphragxn Bracing Interaction
89