CE500 Report Review KSH and Dr Fell 2015-11-30 -

LATERAL STRENGTH OF CORRUGATED WALLS WITH TRAPEZOIDAL GEOMETRY
OF TRANSOCEANIC SHIPPING CONTAINERS
A Project
Presented to the faculty of the Department of Civil Engineering
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
in
Civil Engineering
(Structural Engineering)
by
Blake J. Dolve
FALL
2015
© 2015
Blake J. Dolve
ALL RIGHTS RESERVED
ii
LATERAL STRENGTH OF CORRUGATED WALLS WITH TRAPEZOIDAL GEOMETRY
OF TRANSOCEANIC SHIPPING CONTAINERS
A Project
by
Blake J. Dolve
Approved by:
__________________________________, Committee Chair
Dr. Benjamin Fell, PE
________________________________, Second Reader
Kimberly Scott-Hallet, SE
____________________________
Date
iii
Student: Blake J. Dolve
I certify that this student has met the requirements for format contained in the University format
manual, and that this project is suitable for shelving in the Library and credit is to be awarded for
the project.
__________________________, Graduate Coordinator
Dr. Eugene Dammel
Department of Civil Engineering
iv
___________________
Date
Abstract
of
LATERAL STRENGTH OF CORRUGATED WALLS WITH TRAPEZOIDAL GEOMETRY
OF TRANSOCEANIC SHIPPING CONTAINERS
by
Blake J. Dolve
Refurbished, transoceanic shipping containers are an option for permanent housing. As a
structural system, the primary lateral force resisting components are corrugated walls with
trapezoidal geometries. Due to the lateral force demands from wind and seismic loads, the shear
capacity of the walls must be investigated. This paper analyzes three conditions of stacked
double wide container homes under lateral loads from wind and seismic in both directions. A
range of seismic parameters is considered in order to simulate difference geographic regions.
Equations describing required shear wall lengths as a function of geographic region are
developed. From this analytical study, the corrugated walls of transoceanic shipping containers
are considered adequate as the lateral force resisting system.
_______________________, Committee Chair
Dr. Benjamin Fell, PE
_______________________
Date
v
ACKNOWLEDGEMENTS
I would like to acknowledge Dr. Ben Fell for his encouragement and guidance throughout this
project. I am truly grateful for his time and advice all throughout my undergraduate and graduate
studies. Also, I would like to acknowledge Kim Scott-Hallet and thank her for agreeing to be my
second reader late in the semester. I am very thankful for her dedication to students, like me, who
keep her long after class has ended asking questions and discussing the structural engineering
profession. I would like to acknowledge my friends for those late nights in the structures lab
trying to wrap our heads around complicated problems or workout a bug in a program. Lastly, I
would like to express my deepest thanks to my family who supported me during the good and
challenging times.
vi
TABLE OF CONTENTS
Page
Acknowledgements .................................................................................................................. vi
List of Tables ........................................................................................................................... ix
List of Figures ........................................................................................................................... x
Chapter
1. INTRODUCTION...………………….………………………………………………….. 1
1.1 Goal of Study ............................................................................................................ 1
1.2 Container Geometry and Main Components............................................................. 2
1.3 Load Path .................................................................................................................. 4
1.4 Connecting Member at Shear Wall Locations .......................................................... 4
1.5 Ideal Shear Wall Locations ....................................................................................... 5
1.6 Container Steel Properties......................................................................................... 5
2. BACKGROUND OF THE STUDY ................................................................................... 6
2.1 Review of Research .................................................................................................. 6
2.2 Shear Buckling .......................................................................................................... 6
2.3 Local Panel Bending Stress...… ................................................................................ 8
2.4 Design Drift Limit ..................................................................................................... 9
2.5 Lateral Shear Strength ............................................................................................. 10
3. ANALYSIS OF THE DATA ............................................................................................ 12
3.1 Lateral Loads........................................................................................................... 12
3.2 Lateral Shear Strength Validation ........................................................................... 13
3.3 Governing Equation ................................................................................................ 13
3.4 Load Analysis ......................................................................................................... 14
vii
4. FINDINGS AND INTERPRETATIONS ......................................................................... 15
4.1 Anchorage and Overturning .................................................................................... 15
4.2 Required Shear Wall Length for Base Unit ............................................................ 15
4.3 Conclusion .............................................................................................................. 18
Appendix A. Excel Calculations .......................................................................................... 19
References ............................................................................................................................... 36
viii
LIST OF TABLES
Tables
Page
1.
Governing base shears and overturning forces for Sacramento, California...….…... 12
2.
Governing base shears and overturning forces for Berkeley, California...…….……13
3.
Shear wall strength based on wall geometry…..…….........…………………………14
4.
Critical shear buckling stress and axial load...…………....…………………………14
5.
Proposed LSW Design Equation per Condition Analyzed....…………………………18
ix
LIST OF FIGURES
Figures
1.
Page
Two unit wide (a) one unit high (2W 1H), (b) two unit high (2W 2H) and (c) three
unit high (2W 3H)…....………………………....…..………………………………. 2
2.
40 foot high-cube..…………………….……..……...……………………………. 2
3.
(a) Roof of shipping container, (b) top rail of shipping container , (c) end posts of
shipping container and (d) wall panel section (mm) of 40 foot and 8 foot wall
panel………………………………………….......……………………….……….... 3
4.
Floor diaphragm showing bottom rail and cold-formed channel in mm ...............…. 3
5.
(a) Shear wall placement at corners and (b) plan view of shear walls along the 40
foot and 16 foot faces ……………...…………………….…………………………. 5
6.
Tension and compressive loading due to lateral forces……..…............……………. 7
7.
General labels of trapezoidal panels …..……………………......….....……..………. 7
8.
Accordion effect of trapezoidal wall panel..………..……….…..….....…………..…. 8
9.
Local buckling of Trapezoidal panel due to axial load……………..……...….…..…. 9
10.
Lateral shear strength of corrugated panel …………………...….......….…..….....….10
11.
Shear wall length for the base unit plotted against SDS along the 40 foot wall .....…. 16
12.
Shear wall length for the base unit plotted against SDS along the 16 foot wall .....…. 17
x
1
1. INTRODUCTION
Transoceanic shipping containers are the primary means of import and export around the world.
Due to transportability, applications for refurbished containers may include temporary offices,
finished with insulation and drywall, for companies in construction as well as privately owned
cabins. In addition, relative affordability of shipping containers coupled with material strength,
resilience to corrosion, and physical design has made them a permanent housing option for small
home design. Although the containers are designed to be stacked under tremendous loading and
transported through rigorous and violent oceanic storms, units designed for permanent housing
must be able to withstand lateral forces after unification by means of either bolting or welding,
without allowance for inter-unit slipping, as a system. The structural integrity of the walls in
consideration of lateral loads must be investigated.
1.1
Goal of Study
This paper will investigate the corrugated shear wall capacity of three conditions of double-wide
concept homes of varying height: single, double, and triple stacked. Lateral loading will include
seismic and wind parameters in both directions. Seismic loading will be further investigated by
considering a range of design spectral response acceleration at short periods, SDS, in order to
simulate various geographic locations. Finally, equations describing required shear wall length as
a function of SDS will be developed and modified for all three conditions in both directions. See
Figure 1 below for illustrations of the loading conditions considered with E/W representing
potential governing loads from seismic or wind, respectively.
2
(a)
(b)
(c)
Figure 1. Two unit wide (a) one unit high (2W 1H), (b) two unit high (2W 2H) and (c) three unit
high (2W 3H).
1.2
Container Geometry and Main Components
The shipping container style analyzed is known as a 40 foot high-cube, which is 40 foot long by 8
foot wide by 9 foot 6 inches tall and is illustrated in Figure 2 with Figures 3 and 4 showing
primary components.
Figure 2. 40 foot high-cube.
3
(a)
(b)
(c)
(d)
Figure 3. (a) Roof of shipping container, (b) top rail of shipping container , (c) end posts of
shipping container and (d) wall panel section (mm) of 40 foot and 8 foot wall panel.
Figure 4. Floor diaphragm showing bottom rail and cold-formed channel in mm.
The roof steel is constructed from cambered, die-stamped, corrugated sheets that are 2 mm thick.
The corrugated wall thickness within 4 feet from end posts is 2 mm and 1.6 mm beyond. The
floor diaphragm is composed of 1⅛” marine-grade plywood over cold-formed steel channels
spaced at 12” on center and welded to the bottom rail. The corrugated walls are continuously
4
welded to bottom rail within the flat portion of the 50 mm flange (see Figure 4). The plywood is
screwed to the channels at regular intervals and to the stepped-flange portion of the bottom rail
(see Figure 4). A refurbished container may utilize 2x4 or cold-formed studs and ceiling joists in
order to provide support for gypsum board as well as any miscellaneous electrical loads. The
dead load ratio between the floor, plus roof and ceiling below, for stacked conditions, to tributary
walls heights per story is conservatively 2.4.
1.3
Load Path
The structural characteristics of stacked shipping container homes include a roof, ceiling, walls,
and a floor diaphragm, which transfers gravity and lateral load to end posts. Gravity forces
transferred by the bottom rail to end posts are aided by increased moment of inertia through
continuously welded, corrugated walls with trapezoidal geometry. Due to the small positive
bending moment and high moment of inertia, it is assumed that deflection is negligible near the
corner posts. The walls are continuously connected to the end posts, which are much stiffer than
the walls; therefore, the majority of gravity loads are assumed to be resisted by the end posts.
Since the containers have a relatively stiff floor diaphragm system, lateral loads may be
transferred to the foundation by footings at the corner posts or along any point of the bottom rail.
1.4
Connecting Member at Shear Wall Locations
In order to provide vertical continuity at shear wall locations, a member connecting stacked units
is required. Since the walls are continuously welded to the bottom rail, the connecting steel
member between units defines the length of shear wall. The member must connect the bottom
rail of the upper unit to the top HSS of the lower unit. Member selection in not address in this
5
report. However, the member types that could be considered are angles or plates of sufficient
thickness to transfer lateral and gravity loads present.
1.5
Ideal Shear Wall Locations
Due to the structural design of shipping containers, ideal shear wall locations would be near the
end posts since there would be smaller reduction of shear strength due to deformations from axial
loading (Figure 5). It is assumed that the 8’ doors provide little lateral resistance. All three
configurations are two unit wide systems and arranged so the containers’ doors open at opposite
ends at each level. In addition, stacked configurations are arranged so the doors align vertically.
Therefore, effects due to torsion maybe considered negligible.
(a)
(b)
Figure 5. (a) Shear wall placement at corners and (b) plan view of shear walls along the 40 foot
and 16 foot faces .
1.6
Container Steel Properties
The steel grade of the corrugated wall panels, and most all steel members of transoceanic
shipping containers, is COR-TEN A. Properties include high resistance to corrosion, weldability,
and a yield strength of 50 ksi.
6
2. BACKGROUND OF THE STUDY
Over the last few decades, extensive research has been conducted on corrugated steel walls in
applications as web members for composite precast bridge systems, deck and roof diaphragms,
and sheathing for wall systems resisting lateral forces.
2.1
Review of Research
Analytical studies of walls with trapezoidal geometry loaded parallel to the corrugated subpanels
have been conducted by Yi et al. and Moon et al. and resulted in modifications and refinement to
the interaction between local and global shear-buckling[1][2] . Local bending stress of the flat
panel portion and stiffness as a function of corrugation geometry were researched by Huang et al.
and Ko et al., respectively, for conditions when loading is perpendicular to corrugated
subpanels[3][4]. Shear strength of corrugated walls aligned both parallel and perpendicular to
applied load were tested and found by Emami et al. to possess a higher initial stiffness, ductility
ratio, and energy dissipation capacity than unstiffened walls on the order of 120%, 140%,
and152%, respectively[5] . Research and testing of corrugated walls aligned parallel to loading
was performed by Stojadinovic et al. and resulted in reevaluation of seismic response parameters;
in consideration of seismic loading and drift limits, this paper will accept recommendations to
consider an R = 5.5, Ωo = 2.5, and Cd = 3.25[6]. This paper will consider critical shear-buckling
stress, local panel bending stress, the force required to reach the drift limit, and lateral shear
strength as possible governing equations.
2.2
Shear Buckling
Possible shear buckling failure of the corrugated shear walls is due to the compressive force from
the lateral loading at roof and floor levels as well as the relatively small axial loading due to
7
deformation of the end posts under dead loading (see Figure 6 below).
V
u
T
C
Figure 6. Tension and compressive loading due to lateral forces.
Equations derived and modified by Timoshenko et al., Yi et al., and Moon et al. analyze
corrugated walls as thin, long strips of steel braced from out of plate buckling by the trapezoidal
geometry[7][1][2]. Figure 7 below illustrates trapezoidal section of corrugated wall labeled with
terms used in strength equations.
Figure 7. General labels of trapezoidal panels.
Timoshenko et al. presents an equation describing local buckling of thin plates:
,
where
5.34
4∗
∗
ʋ
∗
,
(1)
is the largest fold width of the corrugated section, which can be , , !#, $ is the wall
height, ʋ is poisson’s ratio, and % is the modulus of elasticity[7]. An equation describing global
buckling has been theorized derived, tested, and modified numerous times. After researched
conducted by Moon et al., the equation has taken the form:
'()
,&
+,*
+,.
/2 ∗ 1
2
8.9:
14 ∗ 1 5 6 7
8
∗
ʋ
∗
(2)
where β is generally taken as a value of 1, and E, t > , , , #, andh have been previously
defined[2]. Due to the geometry of the trapezoidal steel walls, the wall may experience a
combination of both local and global failures when loaded parallel to corrugated panels. The
shear interaction equation presented by Yi et al. is described by:
,C
where
,C
1
D.E,F
D.E,G
4
(3)
is the critical buckling load interaction between local and global failure modes for
shear buckling of trapezoidal walls orientated parallel to anticipated loads[1].
2.3
Local Panel Bending Stress
Due to the trapezoidal geometry of the walls, the critical buckling load described Timoshenko et
al. and Moon et al. can never be fully reached when loaded perpendicular to the corrugated
subpanels. This condition has been coined the “accordion” effect. As the subpanel is compressed
(see Figure 8), the diagonal subpanels crumple and induce a local bending moment on the flat
subpanels.
N
N
Figure 8. Accordion effect of trapezoidal wall panel. Ko et al.
9
Equations evaluating the local flexural bending stress of the flat portion of the subpanel were
studied and modified by Ko et al. and determined to equal:
IJ
I
I 5I
I
Q
I
0.2
ʋ
1
M
Q NO
4'
|L| 8.':
8. :
M |L| 8.P
8.
NO
1 4
RJ '
R 2
8.(
Q
ʋ
1
(4)
(5)
M
RJ '
4
N
'
R 2
Q O
(6)
where TU is the area of the top flange of the wall, V is the ratio of corrugated web height to full
wall height, which is different than 1 in application for precast box-girder bridges with corrugated
steel web systems, and other variables defined previously[4]. Axial force, N, may be factored out
and solved by setting equal to the allowable bending stress:
IJ
IW /ΩJ
(7)
where ΩJ equals 1.67.
2.4
Design Drift Limit
The bending and small axial deformation of subpanels induces a spring-like action resisting axial
force directed perpendicular to the corrugated orientation. Derived from Castigliano’s theorem,
Huang et al. worked with the following equations relating the stiffness of a panel defined by
Figure 9 below.
Figure 9. Local buckling of Trapezoidal panel due to axial load. Huang et al.
10
ZM
Z`
Zd
Q
(8)
JR [ \]^ _
Q
a
[
Q
JR
a
ef
eg
(9)
^bc _
(10)
where ZM and Z` are the axial stiffness from axial loading and bending, respectively, on the
subpanel defined by Figure 9[3]. A nominal shear strength can be determined by multiplying the
trapezoidal panel’s geometric stiffness, given by equation (10), by maximum story drift
prescribed by ASCE 7-10, see equation (11), divided by the displacement amplification factor[8]:
∆
8.8 : Qi
jk
(11)
where $[l is the story height below the level of interest. The required force is then calculated by:
m
2.5
∆ Zd
(12)
Lateral Shear Strength
The strength of a steel shear panel was presented by Sabouri-Ghomi et al. and applied by Emami
et al. to trapezoidal walls with corrugations orientated perpendicular to anticipated load (see
Figure 10 below)[9][5].
νu
νn
/
Figure 10. Lateral shear strength of corrugated panel.
11
The equation describing the lateral shear strength is expressed below:
mop
qr
0.5 ∗ s W sin 2v
(13)
where mop is the shear strength of the panel, q is the wall length, r is the thickness of the panel,
s W is the tension field stress, and v is the angle measured from horizontal of the tension field.
Additional strength due to the frame is neglected for conservatism. After rearranging the Von
Mises yield criterion, Sabouri-Ghomi et al. determined that the tension field yield is a function of
the following[9]:
3
3
s W sin 2v
s W 5 IW
0
(14)
Due to the accordion effect, the critical buckling stress is negligible, which indicates that the
tension field stress may be taken as the yield stress assuming that the members surrounding the
panel are stiff enough for development of the tension field. Therefore, the nominal shear stress
for a trapezoidal panel orientated perpendicular to applied loading is then:
wx
qr 0.5 ∗ IW sin 2v
(15)
In general, theta is taken between 35o and 55o; however, the angle will be determined from the
actual length and height of walls analyzed since previous research used walls with aspect ratios
closer to or less than 1[9], which would validate the prescribed theta value range. Since the
trapezoidal subpanels experience buckling, it is assumed that only the flat portions of panel
provide resistance for shear, which alters equation 15 into the form:
wx
q′r 0.5 ∗ IW sin 2v
where q′ is equal to the flat portions, terms
wall required.
(16)
and
are defined in Figure 7, of the total length of
12
3. ANALYSIS OF THE DATA
For analysis of trapezoidal shear walls, three different configurations of stacked shipping
containers were considered. A two unit wide system was held constant while the height varied
from one to three stacked directly on top of each other. A dead load per floor was determined for
seismic dead load in order for both wind and earthquake lateral analyses to be performed.
3.1
Lateral Loads
Lateral wind and seismic forces are determined from procedures outlined in ASCE 7-10[8]. A
wind velocity of 110 mph consistent with a risk category II 3-second gust geographically located
along the west coast of the United States of America is considered in analysis. A range of
seismic design parameters, which vary geographically, is selected for analysis in order to simulate
different regions used for permanent housing. Tabulated below in Tables 1 and 2 are the
governing base shear and overturning forces at the lowest unit, for all three conditions in both
directions for a geographic location corresponding to Sacramento and Berkeley, California.
Table 1. Governing base shears and overturning forces for Sacramento, California.
2W
1H
2W
2H
2W
3H
V#
Face Governing
40'
Wind
2,452
16'
Seismic
1,810
40'
Wind
7,502
16'
Seismic
5,030
40'
Wind
10,627
16'
Seismic
8,249
Vu #
1,471
1,267
4,501
3,521
6,376
5,774
# SW
2
4
2
4
2
4
Vu # at SW
736
317
2,251
880
3,188
1,444
LSW,Req’d in
17
19
42
47
75
78
Pu # at SW
3,970
1,859
9,942
5,224
16,334
9,351
13
Table 2. Governing base shears and overturning forces for Berkeley, California.
2W
1H
2W
2H
2W
3H
3.2
V#
Face Governing
40'
Seismic
3,192
16'
Seismic
3,192
40'
Seismic
8,870
16'
Seismic
8,870
40'
Seismic
14,549
16'
Seismic
14,549
Vu #
2,235
2,235
6,209
6,209
10,184
10,184
# SW
2
4
2
4
2
4
Vu # at SW
1,117
559
3,105
1,552
5,092
2,546
LSW,Req’d in
22
23
55
58
95
96
Pu # at SW
5,862
3,113
14,418
8,092
25,111
14,183
Lateral Shear Strength Validation
Validation of equation (16) was accomplished through comparison of proprietary products.
Corrugated geometry and wall heights for 12” and 24” long shear wall panels manufactured by
Simpson Strongtie were inputted into the strength equations listed above[10]. It was found that the
controlling nominal strength equation was greater, by a factor less than 2.35, for shear values
listed in Simpson Strongtie’s catalog. According to the Steel Deck Institute’s Design manual,
allowable strength design factors are 2.5 and 2.35 for seismic and wind, respectively[11]. It is
assumed that testing of propriety products yield values greater than those in the design manual
and that the equations presented above are reasonably accurate for design of trapezoidal shear
walls.
3.3
Governing Equation
Out of the three lateral design equations describing drift limit (4), local panel bending stress (12),
and shear strength (16), is was found that shear strength governed (see Table 3 below). The shear
strength within 4 feet from end posts for both 40 foot and 8 foot corrugated wall geometries are
listed in Table 3 below. An allowable strength design factor of 2.5 and 2.35 are used for seismic
and wind, respectively[11]. The capacity of the 8 foot wall geometry is greater than the 40 foot
wall geometry because the corrugations are tighter, there is more flat length of steel, per foot.
14
Table 3. Shear wall strength based on wall geometry.
40' Wall
8' Walls
Vn #
Vn/ΩW #
Vn/ΩE #
2,760
4,744
1,175
2,019
1,104
1,898
However, after consideration of overturning compressive loads, it was determined that shear
buckling of the panel ultimately governed in both directions for all conditions. Table 4 below
lists the critical buckling stress and axial loading per 12 inches for both wall geometries. An
allowable strength design factor of 1.67 was used for axial loading. The 40 foot wall row has two
columns per field; this is because the corrugated wall thickness within the first 4 feet, represented
by the first set of data per field, is 2 mm as opposed to 1.6 mm beyond.
Table 4. Critical shear buckling stress and axial load.
16' Wall
40' Wall
3.4
τcr, I (ksi)
Pn # (per 12”)
Pn /ΩC # (per 12”)
9.52
4,496
2,692
6.09
5.36
3,823
2,878
2,289
1,723
Load Analysis
Required Lengths per value of SDS were determined by increasing trapezoidal wall length in one
inch increments until seismic demand load per SDS was exceeded. Since the ratio of flat regions
of corrugated wall per foot is less than 1, data acquired from small increments appear as a partial
stepwise function. Values for SDS were increased incrementally by 0.10 until a maximum of 1.25
was reached.
15
4. FINDINGS AND INTERPRETATIONS
Trapezoidal steel shear walls with geometry matching transoceanic shipping containers were
analyzed analytically in this paper. Three conditions were tested, all two unit wide with heights
varying from one to three units, for both 40 foot and 8 foot trapezoidal wall geometries. Lateral
loads were determined following ASCE 7-10 guidelines for wind and seismic. Selected strength
equations for shear design are reasonable considering guidelines in the Steel Deck Institute
Diaphragm Design Manual and comparison to proprietary product design tables.
4.1
Anchorage and Overturning
Design of wall and end post foundations were not addressed in this report. Prudent design would
include footings under each post and a continuous footing with anchors attached to the bottom
rails on all walls spaced at regular intervals. The floor system of the lowest unit is a relatively
stiff diaphragm able to transfer lateral loads from one end to the other. Therefore, a system where
the primary connection to the foundation occurs at the end posts, depending on the geographic
location as well as overall building height and width, may be determined adequate. Since the
base shear for the worst possible case was relatively small (see Table 2), anchorage requirements
for global stability requirements will most likely control for conditions where the total building
height is larger than the shortest side.
4.2
Required Shear Wall Length for Base Unit
As presented in Figures 11 and 12 below, the required length of shear walls for the base units
along the 40 foot and 16 foot walls was computed as a function of seismic loading. Design per
wind loading governs approximately within the region left of the first data points. For shear walls
along the 40 foot and 16 foot faces, this correlated to an SDS value approximately between 0.20 to
16
0.35 and 0.45 to 0.75, respectively. The maximum value of SDS was arbitrarily selected past a
value of 1.0 in order to acquire better equations for linear interpolation. Per ASCE 7-10, section
12.8.1.3 allows for calculation of SDS based on a Ss value of 1.5 for regular structures five stories
or less, which corresponds to an SDS of 1.0 for seismic site classes A through D[8]. The vertical
lines on the represent the regions for Sacramento, SDS = 0.567, and Berkeley, which represents the
maximum SDS of 1.0[12].
Shear Wall Length Along 40 foot walls Vs. SDS
Shear Wall Length (Feet)
10.00
2W3H
9.00
2W2H
8.00
2W1H
7.00
Sacramento
6.00
Berkeley
y = 4.36x + 3.57
R² = 0.98
5.00
y = 2.68x + 1.88
R² = 0.99
4.00
3.00
2.00
y = 1.05x + 0.75
R² = 0.98
1.00
0.00
0.000
0.250
0.500
0.750
1.000
1.250
1.500
SDS
Figure 11. Shear wall length for the base unit plotted against SDS along the 40 foot wall.
17
Shear Wall Length (Feet)
Shear Wall Length Along 16 foot walls Vs. SDS
10.00
2W1H
9.00
2W2H
8.00
2W1H
7.00
Sacramento
6.00
y = 3.38x + 4.63
R² = 0.96
Berkeley
5.00
y = 2.03x + 2.75
R² = 0.99
4.00
3.00
2.00
y = 0.82x + 1.13
R² = 0.95
1.00
0.00
0.000
0.250
0.500
0.750
1.000
1.250
1.500
SDS
Figure 12. Shear wall length for the base unit plotted against SDS along the 16 foot wall.
Per analytical analysis, required shear wall length for corrugated walls with trapezoidal
configuration varies linearly per SDS based on geographic location, which is anticipated. The
maximum required shear wall length for the system per SDS equal to 1 occurs along the 40 foot
wall. The shear wall length along the 16 foot wall is capped at 8 foot since there are end posts at
that location and shear buckling controls up to that length (see page 35 of Appendix A).
Therefore, the three systems provide an adequate length of wall in both directions. Based on
equations presented in Figures 11and 12, proposed shear wall length design equations, for the
systems analyzed in this paper, are presented in Table 5 below. Prudent design would round up
to the nearest foot or half a foot.
18
Table 5. Proposed LSW Design Equation per Condition Analyzed.
Condition
Proposed LSW Design Equation
2W1H—Along 40 foot Face
qz{
1.05|2[
0.75Ir,0.25 ≤ |2[ ≤ 1.0
(15)
2W2H—Along 40 foot Face
qz{
2.68|2[
1.88Ir,0.25 ≤ |2[ ≤ 1.0
(16)
2W3H—Along 40 foot Face
qz{
4.36|2[
3.57Ir,0.25 ≤ |2[ ≤ 1.0
(17)
2W1H—Along 16 foot Face
qz{
0.82|2[
1.13Ir,0.45 ≤ |2[ ≤ 1.0
(18)
2W2H—Along 16 foot Face
qz{
2.03
2W3H—Along 16 foot Face
qz{
3.38|2[
4.3
2.75Ir,0.45 ≤ |2[ ≤ 1.0
(19)
4.63Ir,0.45 ≤ |2[ ≤ 0.95
(20)
Conclusion
The base unit shear wall lengths required to resist wind or seismic loading for the trapezoidal
siding of transoceanic shipping containers were analytically investigated in this paper.
Conditions for a permanent residence consisting of two units side by side stacked one to three
units high were considered. The capacity of the corrugated walls were determined with
consideration of shear buckling, local panel bending stresses, the lateral force required to reach
drift limits, and lateral shear strength. The governing design equation with consideration of
strength reduction factors, used in engineering practice, for wind and seismic was compared
against proprietary products of similar design and found to be acceptable. Required shear wall
lengths along both 16 foot and 40 foot walls of the base unit were plotted against varying values
of SDS. Equations describing required shear wall lengths for the base unit were developed from
varying values of SDS. The trapezoidal, exterior walls of transoceanic shipping containers provide
adequate resistance to lateral forces within reasonable wall lengths required.
19
Appendix A. Excel Calculations
20
Design Criteria
Dimensions
Plan
40
16
Plan Area
Side Wall Height
ft long
ft wide
640 sf
8.67 ft
Height to Top Rail
9.5 ft
Perimeter
114 ft
Height to each level
1H
2H
3H
Area
40'*9.5'
16'*9.5'
Loads
DL:
9.5
19
28.5
380 sf
152 sf
Roof
Floor
Side Wall
Potential Metal Roof
Top (Ceiling Steel Plate)
5/8" Gypsum Board
Future mechanical (Solar)
Mechical/Electrical
Misc Framing
5/8" Hardwood Flooring (Finish)
1 1/4" Marine hardwood flooring
C-Channel Framing
Mechical/Electrical
Misc
Top Rail
Perimeter Wall (8'-8")
Bottom Rail
1.5
5.1
2.8
3
2
1.6
2.5
4
5
2
1.5
3.5
33.2
3.8
Psf
Psf
Psf
psf
psf
psf
psf
psf
psf
psf
psf
plf
plf
plf
LL:
Roof
Floor
15psf
+
40psf
=
20 psf
55 psf
21
Seismic
(Berkeley)
USGS
Ss
2.306
S1
0.958
SDS
1.00
Sd1
0.958
Seismic Design Coefficients [x]
R
5.5
Ω0
Cd
Approximate Period
T
2.5
3.25
0.247
Seismic Response Coefficient
Cs
0.182
Cs, max
Cs, min
Governs
0.706
0.044
Seismic Dead Load: Roof and Floors Above
Roof
13,839 #
Floors
23,793 #
Seismic Dead Load of Walls (one wall--Multiply by appropriate integer)
40 ft
2,656 #
16 ft
1,062 #
22
ƒ
Wind
0.00256ˆ‰ ˆ‰ ˆ2 w
q=
q (9.5')
q (19')
q (28.5')
‚
ƒ„…† 5 ƒ‡ „…†‡
40
Kzt
Kd
V
Kz (0-15ft)
Kz (15-20ft)
Kz (25-30ft)
26.33 Kz
15.01
16.06
17.90
1
0.85
110 mph
0.57
0.61
0.68
G
Cp, windward
Cp,
0.85
0.8
leeward
-0.28
Cp,
16
leeward
0.5
Cp,
sidewall
GCpi
Side Note:
-0.7
±0.18
|0.85 ∗ 0.8 5 50.18 | 0.86 Š|-0.5*0.85-0.18|=|-0.61|
Therefore, use postive pressures
Pressure
Windward
0-15'
1520'
2530'
12.91
@9.5'
13.81
@19'
15.40
@28.5'
23
Unit: 2W 1H
Seismic
Seismic Dead
17,557
#
V=
3,192 @9.5'
h (ft)
whk
Cvx
Fx #
9.5
166,795
1.00
3,192
h (ft)
whk
Cvx
Fx #
9.5
166,795
1.00
3,192
Distribution of Forces
40
Level
ft
Face
Roof
16
Level
ft
Face
Wind
40
ft
Face
16
ft
Face
Total
DL #
17,557
Roof
Total
DL #
17,557
Level
Roof
Fx #
2,452
∑Fx #
2,452
Level
Roof
Fx #
981
∑Fx #
981
∑Fx #
3,192
∑Fx #
3,192
Governs
Governs
24
Unit: 2W 2H
Seismic
Seismic Dead Load
48,787 #
Distribution of Forces
Total
40
Level
h (ft)
DL #
ft
Roof
17,557
19
Face
2nd Floor 31,230
9.5
∑
48,787
Total
16
Level
h (ft)
DL #
ft
Roof
17,557
19
Face
2nd Floor 31,230
9.5
∑
48,787
Wind
40
Level
Fx #
∑Fx #
ft
Roof
2,597 2,597
Face
2nd Floor 4,905 7,502
∑
7,502
16
Level
Fx #
∑Fx #
ft
Roof
1,104 1,104
Face
2nd Floor 2,084 3,188
∑
3,188
V=
8,870 #
whk
Cvx
Fx #
333,591
296,685
630,276
0.53
0.47
1.00
4,695
4,176
8,870
whk
Cvx
Fx #
333,591
296,685
630,276
0.53
0.47
1.00
4,695
4,176
8,870
∑Fx #
4,695
8,870
Governs
∑Fx #
4,695
8,870
Governs
25
Unit: 2W 3H
Seismic
Seismic Dead Load
80,017 #
Distribution of Forces
40
Level
Total DL #
ft
Roof
17,557
Face
3rd Floor
31,230
2nd Floor
31,230
∑
80,017
16
Level
Total DL #
ft
Roof
17,557
Face
3rd Floor
31,230
2nd Floor
31,230
∑
80,017
Wind
40
Level
Fx #
ft
Face
Roof
3rd Floor
2,926
5,249
2nd Floor
∑
16
Level
ft
Face
Roof
3rd Floor
2nd Floor
∑
2,452
10,627
Fx #
1,170
2,100
981
4,251
V=
h (ft)
28.5
19
9.5
h (ft)
28.5
19
9.5
∑Fx
#
2,926
8,174
10,62
7
∑Fx
#
1,170
3,270
4,251
whk
500,386
593,370
296,685
1,390,441
whk
500,386
593,370
296,685
1,390,441
14,549
Cvx
0.36
0.43
0.21
1.00
Cvx
0.36
0.43
0.21
1.00
#
Fx #
5,236
6,209
3,104
14,549
Fx #
5,236
6,209
3,104
14,549
∑Fx #
5,236
11,444
14,549
Governs
∑Fx #
5,236
11,444
14,549
Governs
26
Unit: 2W 1H
Governing
Loads per
ASD
0.6 W;
0.7 E
40
ft
Face
Level
Roof
Wind
1,471
Seismic
2,235
Governing
Seismic
2,235 #
16
ft
Face
Level
Roof
Wind
589
Seismic
2,235
Governing
Seismic
2,235 #
SW1 =
559
SW Loads @ base
#
SW2 =
1,117
#
27
Unit: 2W 2H
Governing
Loads per
ASD
0.6 W;
0.7 E
40
Level
Wind
Seismic
Governing
ft
Roof
1,558
3,286
Seismic
3,286 #
Face
2nd
4,501
6,209
Seismic
6,209 #
16
ft
Level
Roof
Wind
662
Seismic
3,286
Governing
Seismic
3,286 #
Face
2nd
1,913
6,209
Seismic
6,209 #
SW1 =
1552
SW Loads @ base
#
SW2 =
3,105
#
28
Unit: 2W 3H
Governing
Loads per
ASD
40
ft
Face
0.7 E
Level
Roof
Wind
1,755
Seismic
3,665
Governing
Seismic
3rd Floor
4,905
8,011
Seismic
2nd
6,376
10,184
Seismic
10,184 #
Level
Roof
Wind
702
Seismic
3,665
Governing
Seismic
3,665 #
3rd Floor
1,962
8,011
Seismic
8,011 #
2nd
2,550
10,184
Seismic
10,184 #
SW1 =
2546
#
16
ft
Face
0.6 W;
3,665 #
8,011
SW Loads @ base
SW2 =
5,092
#
29
Design Wall Geometries and Properties, 40'
General Properties and Criteria
Per steel deck institute design manual
E
W
φ
Ω
0.65
2.5
0.7
2.35
a=
c=
b=
d=
θ=
2.8346
3.0292
2.6772
1.4173
27.90
h=
t=
104 in
0.0630 in
Af =
1.15678 sq in
Es =
v=
29000 ksi
0.3
Fy =
50 ksi
Spec
2.8346
note: t varies from 2mm for the first 4 feet to 1.6mm past 4
feet.
Although a t of 1.6mm is used to compare equations, 2mm is
used when L of the shear wall is less than or equal to 4 feet.
in
in
in
in
degrees
SST 12" SST24
1.3125 2.84375
3.0292
3.2944 3.6239
2.25 2.5625
2.6772
1.4173
2.4063 2.5625
27.90
46.92 45.00
104
120 120
0.0630
0.1345 0.1345
Simpson takes into consideration the
following limit states: drift, tension and
compression, flexure, anchor rod
tension, and concrete or wood bearing
stress. The single-story wall tables, the
limit state is allowable shear.
30
Shear Wall Capacity, 40'
Wall L
12 in
Design Per
ASD Ω =
, for comparison
2.5
Local Accordion Effect---Local bending stresses
Ko et
al.
2013
fbt =
6.08 *N sq in
1/Ω Na =
5,358 plf
=
1/1.67
x
Does NOT Govern
Shear Yield
Emami et al. (2012) & Sabouri-Ghomi et al (2005)
θtension
83.42 deg
field
τy =
Vn =
a per ft
Vn, #a =
1/Ω Vn, #a =
1/Ω Vn, wall =
5.69
359
6.1575
2,208
883
ke,panel =
Governs
883 #, wall
Allowable Drift
L. Huang et al.
2004
kN =
351.3 k/in/in
kM =
ksi
# per in
in
plf
plf
&
ASCE 7-10
0.3 k/in/in
0.31 k/in/in
ke,panel =
33 k/in
# Panel
2
ASCE 7-10 Table 12.12-1
h (ft)
∆limit in
8.67
0.22
Shear Buckling
Moon et al.
∆limit/Cd in
0.07
Vallow #
4,719 Does NOT Govern
2008
kL =
9.34
kG =
630.11
kI =
596.00
τcr, I =
5.36 ksi
Critical buckling strength equals 0.5L x tw when comparing to chord force Pu
50
ksi
31
Shear Equation Calibration
SSW12x
Axial Load = 1000 #
h (ft)
Vn, #a #
SST Va, E #
ΩE, Cal.
SST Va, W #
ΩW, Cal.
7
1,899
955
1.99
1,215
1.56
7.4
8
9
10
1,798
1,666
1,483
1,336
870
775
660
570
2.07
2.15
2.25
2.34
1,105
985
840
725
1.63
1.69
1.77
1.84
h (ft)
Vn, #a #
SST Va, E #
ΩE, Cal.
SST Va, W #
ΩW, Cal.
7
7.4
1,899
1,798
955
870
1.99
2.07
1,095
970
1.73
1.85
8
1,666
775
2.15
865
1.93
9
1,483
660
2.25
705
2.10
10
1,336
570
2.34
570
2.34
SSW12x
Axial Load = 7500 #
h (ft)
Vn, #a #
SST Va, E #
ΩE, Cal.
SST Va, W #
ΩW, Cal.
7
7.4
8
9
10
1,899
1,798
1,666
1,483
1,336
890
750
665
505
360
2.13
2.40
2.51
2.94
3.71
1,095
970
865
705
570
1.73
1.85
1.93
2.10
2.34
SSW12x
Axial Load = 4000 #
Omega per design manual is greater than SST. This is expected. Equation describing shear
capacity is OK
φ
Ω
0.65
2.5
E
0.7
2.35
W
32
Shear Equation Validation
SSW24x
Axial Load = 1000 #
h (ft)
Vn, #a #
12,14
7
2
11,55
7.4
3
10,76
8
3
9
9,654
10
8,745
SSW24x
Axial Load = 4000 #
h (ft)
Vn, #a #
SST Va, E #
ΩE, Cal.
SST Va, W #
ΩW, Cal.
5,495
2.21
5,730
2.12
5,330
2.17
5,450
2.12
4,865
4,285
3,835
2.21
2.25
2.28
5,105
4,605
4,205
2.11
2.10
2.08
SST Va, E #
ΩE, Cal.
SST Va, W #
ΩW, Cal.
7
7.4
8
9
12,142
11,553
10,763
9,654
5,495
5,330
4,865
4,285
2.21
2.17
2.21
2.25
5,730
5,450
5,105
4,605
2.12
2.12
2.11
2.10
10
8,745
3,835
2.28
4,205
2.08
SSW24x
Axial Load = 7500 #
h (ft)
Vn, #a #
SST Va, E #
ΩE, Cal.
SST Va, W #
ΩW, Cal.
7
7.4
8
9
10
12,142
11,553
10,763
9,654
8,745
5,495
5,330
4,865
4,285
3,790
2.21
2.17
2.21
2.25
2.31
5,730
5,450
5,105
4,480
3,790
2.12
2.12
2.11
2.15
2.31
Omega per design manual is greater than SST. This is expected. Equation describing shear
capacity is OK
E
W
φ
0.65
0.7
Ω
2.5
2.35
33
Design Wall Geometries and Properties, 16 ft
General Properties and Criteria
Per steel deck institute design manual
E
W
φ
0.65
0.7
Ω
2.5
2.35
a=
c=
b=
d=
θ=
h=
t=
Af =
Es =
v=
Fy =
Spec
4.3307
1.9301
0.7087
1.7953
68.46
104
0.0787
1.55926
29000
0.3
50
SST 12"
in
in
in
in
degrees
in
in
sq in
ksi
ksi
SST24
4.3307
1.3125 2.84375
1.9301
3.2944
3.6239
0.7087
2.25
2.5625
1.7953
2.4063
2.5625
68.46
46.92
45.00
104
120
120
0.0787
0.1345
0.1345
SST takes into consideration
the following limit states:
drift, tension and
compression, flexure, anchor
red tension, and concrete or
wood bearing stress. The
single-story wall tables, the
limit state is allowable shear.
34
Shear Wall Capacity
16 ft
Wall L
12 in
Design Per
ASD Ω =
2.35
Local Accordion Effect---Local bending stresses
Ko et
al.
2013
fbt =
4.35 *N sq in
=
1/1.67
1/Ω Na =
8,188 plf
Does NOT Govern
x
50 ksi
Shear Yield
Emami et al. (2012) & Sabouri-Ghomi et al (2005)
θtension
83.42 deg
field
5.69 ksi
τy =
Vn =
448 # per in
a per ft
10.5827
in
4,744 plf
Vn, #a =
1/Ω Vn, #a =
2,019 plf
Governs
1/Ω Vn, wall =
2,019 #, wall
Allowable Drift
L. Huang et al.
kN =
kM =
ke,panel =
ke,panel =
# Panel
497.4
0.3
0.3
31
2
2004
ASCE 7-10
k/in/in
k/in/in
k/in/in
k/in
ASCE 7-10 Table 12.12-1
h (ft)
Cd
8.67
&
3.25
driftlimit in
driftlimit/Cd in
0.22
0.07
Vallow #
Does NOT
4,857 Govern
Shear Buckling
Moon et al.
2008
kL =
9.34
kG =
694.56
kI =
677.22
τcr, I =
9.52 ksi
Critical buckling strength equals 0.5L x tw when comparing to chord force Pu
35
Required Shear Wall Length For Base Unit
Design Per
Shear Yield
Emami et al. (2012) & Sabouri-Ghomi et al (2005) ---Vn
Shear
Buckling
Moon et al. (2008) ---Pn
SDS = 1.0
2W
1H
Wall
40
16
2W
2H
Wall
40
16
2W
3H
Wall
40
16
Seismic
ASD
Ωv
2.5
Seismic
2.5
22
23
ASD
Ωv
2.5
2.5
SWL
in
55
58
Vn /Ω
Vu #
#
18,890 1,552
37,872 3,105
ASD
Ωc
1.67
1.67
Pn /Ωc
Pu #
#
8,214 8,000
14,833 13,607
ASD
Ωv
2.5
2.5
SWL
in
95
96
Vn /Ω
Vu #
#
39,526 2,546
66,450 5,092
ASD
Ωc
1.67
1.67
Pn /Ωc
Pu #
#
14,187 13,842
24,552 25,111
Governing
Governing
Seismic
Seismic
Governing
Seismic
Seismic
SWL
in
Vn /Ω
#
3,600
7,890
559
ASD
Ωc
1.67
Pn /Ωc
#
3,286
3,113
1,117
1.67
5,882
5,447
Vu #
Note: For 16' wall, end posts are present when shear wall is 8' long. At that point,
Shear controls
Required Shear Wall Lengths (Feet) Per SDS
SDS
1H 40'
2H 40'
3H 40'
1H 8'
2H 8'
0.250
0.350
0.450
0.550
0.650
0.750
0.850
0.950
1.050
1.150
1.250
0.92
1.08
1.25
1.33
1.50
1.58
1.67
1.75
1.83
1.92
2.00
2.42
2.75
3.08
3.42
3.67
4.00
4.25
4.50
4.67
4.92
5.08
4.25
5.00
5.67
6.17
6.58
7.00
7.42
7.75
8.17
8.50
8.75
1.58
1.58
1.58
1.58
1.58
1.67
1.83
1.92
2.00
2.08
2.17
3.75
3.75
3.75
3.92
4.00
4.17
4.42
4.67
4.92
5.08
5.33
3H
8'
6.33
6.33
6.33
6.42
6.67
7.08
7.58
7.92
8.00
8.00
8.00
Pu #
36
References
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corrugated steel shear walls. Engineering Structurals 2013; 48: 750-762.
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International Specialty Conference on Cold-Forme Steel Structures 2008; Paper 5.
7. Timoshenko SP, Gere JM. Theory of elastic stability. 2nd ed. New York: McGraw-Hill; 1961.
8. ASCE. Minimum Design Loads for Buildings and Other Structures. American society of
Civil Engineers. Structural Engineering Institute 2013; 3rd Ed.
9. Sbouri-Ghomi S, ventura CE, Kharrazi MHK. Shear analysis and design of ductile steel plate
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C-L-SW14.
10. Steel Deck Institute (SDI). (2004). Diaphragm design manual, 3rd Ed., SDI, Canton, OH.
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<http://earthquake.usgs.gov/designmaps/us/application.php>