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Elastic Buckling Studies of Thin Plates and
Cold-Formed Steel Members in Shear
by
Rakesh Timappa Naik
Report submitted to the Faculty of Virginia Polytechnic Institute and State
University in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
In
Civil Engineering
APPROVED
Dr. Christopher D. Moen, Chairperson
Dr. W. Samuel Easterling Dr. Finley A. Charney
December 2010
Blacksburg, Virginia
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i
Elastic Buckling Studies of Thin Plates and Cold-Formed SteelMembers in Shear
Rakesh Timappa Naik
ABSTRACT
Shell finite element elastic buckling studies of thin plates and full cold-formed
steel members are conducted which lead to finite element guidelines for modeling
thin-walled members in shear. The influence of cross-section connectivity on shear
buckling stress for industry standard cold-formed steel cross-sections is
summarized. A shear buckling energy solution is derived including rotational
springs which can be used to quantify the influence of cross-section connectivity
on the shear buckling stress. Finite element eigen-buckling analysis of plates with
spring stiffness simulating the effect of cross-section connectivity are conducted to
develop an expression for plate buckling coefficient. The research effort is the
first step in development of a simplified method for predicting the critical elastic
buckling load of cold-formed steel members in shear including cross-section
connectivity. Hand methods for predicting shear buckling which include cross-
section connectivity are needed to support the extension of the American Iron and
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Steel Institutes (AISI) Direct Strength method to cold-formed steel members in
shear.
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To my parents
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Acknowledgements
This report would not have been possible without the guidance and support
of many people. First and foremost, I owe my deepest gratitude to my advisor, Dr.
Christopher Moen for supporting me throughout my thesis with his patience and
knowledge. I am grateful to him for his constant encouragement and giving me
an opportunity to work with CFS.
I would like to thank Dr. Charney and Dr. Easterling for graciously
agreeing to be a part of my committee and for enriching my learning experience at
Virginia Tech. I really enjoyed your classes.
I would like to take this opportunity to thank all the people who helped me
in this research, specially the following graduate students: Karthik Ganesan, Amey
Bapat, Rohan Talwalkar, Behrooz Soorori Rad, Maninder Bajwa, Adrian Tola,
Leonardo Hasbun, Fae Garstang and Vathana Poev.
A special thank you goes out to Vidula Bhadkamkar for all the continuous
support. Thank you Kalyani Tipnis, Rohit Kota, Shambhavi Reddy, Mandar
Waghmare, Kunal Mudgal, Gaurav Mehta, Gokul Kamath and Dhawal Ashar for
being great friends and making my stay at Blacksburg so memorable.
Finally, I would like to thank my father, Timappa Naik for encouraging me
to take up higher studies and supporting me throughout my way. A heartfelt thanks
to my mother, Sharada Naik and brother, Prasanna for their unconditional love and
support.
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TABLE OF CONTENTS
List of Figures ...................................................................................................... viiChapter 1 Introduction ......................................................................................... 1
1.1 Cold-formed steel - History and Uses ........................................................ 11.2 Direct Strength Method (DSM).................................................................. 31.3 Research Motivation .................................................................................. 7
Chapter 2 Classical solution for shear buckling .............................................. 12Chapter 3 Finite element modeling guidelines for thin plates in shear.......... 28
3.1 Summary of ABAQUS thin-shell elements ............................................. 293.2 Loading and Boundary conditions ........................................................... 323.3 Summary of ABAQUS thin-shell elements ............................................. 34
Chapter 4 shear elastic buckling Studies on channel sections ........................ 404.1 Finite Element Modeling Assumptions .................................................... 414.2 Loading and Boundary Conditions .......................................................... 42
4.3 Elastic Buckling Analyses ........................................................................ 43
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4.4 Extension of the classical shear buckling equation for plates withrotational springs ...................................................................................... 46
4.5 Verification by Finite-Element Analysis .................................................. 504.6 Design Expression for Plate Buckling ..................................................... 55
Chapter 5 Design Implementation ..................................................................... 585.1 Rotational stiffness ................................................................................... 595.2 In plane bending stiffness ......................................................................... 615.3 Verification by Finite-Element Analysis .................................................. 62
Chapter 6 Conclusions and Future Work ......................................................... 676.1 Conclusions .............................................................................................. 676.2 Recommendations for future research ...................................................... 69
REFERENCES .................................................................................................... 70APPENDIX A ...................................................................................................... 73APPENDIX B ...................................................................................................... 77
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LIST OF FIGURES
Figure 1-1: Elastic Buckling curve for a cold-formed steel beam .......................... 4Figure 1-2: DSM Distortional buckling design curve ............................................. 6Figure 1-3: DSM local buckling design curve ........................................................ 6Figure 1-4: DSM Lateral-torsional buckling design curve ..................................... 7Figure 1-5 : Web is treated as a simply-supported plate in design ......................... 9Figure 2-1: Plate coordinate system and dimension notation ............................... 13Figure 2-3 : (a) Symmetric buckle mode (b) anti-symmetric buckle mode ........ 20Figure 2-4: Choice of Equations (a) m+n even (b) m+n odd ................................ 22Figure 2-5 : Comparison between solutions using 10 and 20 Fourier series terms.26 Figure 3-1: a) S4/ S4R shell element (b) S9R5 shell element .............................. 29Figure 3-2: a) Four point integration rule (b) One point integration rule .......... 30Figure 3-3 : A plate element with corner nodes showing a midsurface. ............... 30Figure 3-4: Plate boundary conditions and loading .............................................. 33Figure 3-5 : Rigid body rotation restrained in ABAQUS ..................................... 34Figure 3-6: Variation in ABAQUS predicted buckling coefficient kv with number
of elements per buckled half-wave .................................................. 36
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Figure 3-7: Variation in ABAQUS predicted buckling coefficient kv with element
aspect ratio, A/B ................................................................................ 37Figure 3-8 : Variation in ABAQUS predicted buckling coefficient kv with plate
aspect ratio, a/b .................................................................................. 39Figure 4-1: (a) Member loading, boundary conditions and dimension notation, (b)
cross-section dimension range where H, B, L, D are out-to-out
dimensions, ris the inside radius and tis the thickness. ................... 42Figure 4-2: (a) SSMA 1200S200-54 web plate and structural stud, (b) SSMA
800S200-54 web plate and structural stud, and (c) SSMA 400S200-54
web plate and structural stud. ............................................................ 44Figure 4-3: Variation in crwithH/B for SSMA sections forL/H=8.0. ................ 45Figure 4-4: Plate coordinate system and dimension notation ............................... 47Figure 4-5:Spring element between node and the ground with 6 degree of freedoms
(SPRING1) ........................................................................................ 51Figure 4-6: Comparison ofkv calculated with energy solution to FE eigen buckling
solution with rotational restraint ........................................................ 52Figure 4-7: Variation in kv with plate aspect ratio a/b and k ............................... 54
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Figure 4-8: Family of curves used to simulate the effect of rotational springs on the
buckling capacity. .............................................................................. 56Figure 5-1 : Buckled shape of an SSMA 800S200-54 section ............................. 59Figure 5-2: ABAQUS boundary conditions and imposed rotation for the web plate
........................................................................................................... 60Figure 5-3: Rotational stiffness of the plate .......................................................... 60Figure 5-4: ABAQUS boundary conditions and imposed rotation for the web plate
........................................................................................................... 61Figure 5-5: In plane bending stiffness of the plate ............................................... 62 Figure 5-6: ABAQUS loading conditions for the plate ........................................ 64Figure 5-7: Variation in kv with plate aspect ratio and krand kt........................ 65Figure 5-8: Variation in kv with varying rotational and in plane bending stiffness
....................................................................................................... 66Figure 5-9: Surface plot fitting the variation in kv with varying rotational and in
plane bending stiffness ...................................................................... 68Figure A-1: Shape change of the Block under the Moment in ideal situation ...... 73Figure A-2: Shape change of a fully integrated first order element under the
Moment ............................................................................................ 74
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Figure A-3: Shape change of a reduced integrated first order element under the
momentM........................................................................................ 76
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CHAPTER 1 INTRODUCTION1.1 Cold-formed steel - History and Uses
Cold-formed steel (CFS) members have been used in buildings, bridges,
storage racks, grain bins, car bodies, railway coaches, highway products,
transmission towers, transmission poles and drainage facilities. The use of cold-
formed steel members in building construction began in the 1850s. In the United
States, the first edition of the Specification for the Design of Light Gage Steel
Structural Members was published by the American Iron and Steel Institute (AISI)
in 1946 (AISI, 1946). In 2001, the first edition of the North American
Specification for the Design of Cold-Formed Steel Structural Members was
developed by a joint effort of the AISI Committee on Specifications, the Canadian
http://en.wikipedia.org/wiki/Grain_binhttp://en.wikipedia.org/wiki/Drainagehttp://en.wikipedia.org/wiki/Canadian_Standards_Associationhttp://en.wikipedia.org/wiki/Canadian_Standards_Associationhttp://en.wikipedia.org/wiki/Canadian_Standards_Associationhttp://en.wikipedia.org/wiki/Drainagehttp://en.wikipedia.org/wiki/Grain_bin7/28/2019 Naik Shearbuckling
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Standards Association (CSA) Technical Committee on Cold-Formed Steel
Structural Members, and Camara Nacional de la Industria del Hierro y del Acero
(CANACERO) in Mexico (AISI, 2001). It included the ASD and LRFD methods
for the United States and Mexico together with the Limit States Design (LSD)
method for Canada. This North American Specification has been accredited by the
American National Standard Institute (ANSI) as an ANSI Standard to supersede
the 1996 AISI Specification and the 1994 CSA Standard. Following the
successfully use of the 2001 edition of the North American Specification for six
years, it was revised and expanded in 2007. This updated specification includes
new and revised design provisions with the additions of the Direct Strength
Method in Appendix 1 and the Second-Order Analysis of structural systems in
Appendix 2. Currently, two strength prediction methods for cold-formed steel
members are available, the traditional Effective Width Method in the main
Specification and the Direct Strength Method. This report concentrates on the
Effective width method.
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1.2 Direct Strength Method (DSM)The Direct Strength Method is used for determining the strength
(resistance) of cold-formed steel members (beams and columns). This method
uses the buckling properties of an entire cross-section to calculate the capacity
(Schafer 2002). This method has the advantage that calculations for complex
sections are very simple, provided elastic buckling solutions are available.
The elastic buckling solutions suitable for use with DSM can be obtained
from the finite strip method for buckling analysis (Cheung and Tham 1998). The
American Iron and Steel Institute has sponsored research that, in part, has lead to
the development of freely available program, CUFSM, which employs the finite
strip method for elastic buckling determination of any cold-formed steel cross-
section. The buckling modes which control the capacity of a cold-formed sectionare assumed to be local buckling, distortional buckling and global (Euler)
buckling. An elastic buckling curve and the corresponding buckling modes for a
channel section, generated using CUFSM are shown in Figure 1-1.
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Figure 1-1: Elastic Buckling curve for a cold-formed steel beam
The elastic buckling solutions obtained from finite strip analysis are used to
calculate the elastic buckling loads and/or moments with the help of DSM
equations. The equations necessary for the calculation of strength of cold-formed
steel columns and beams are provided in the Appendix 1 of the AISI specifications
and also given in Figure 1-2, Figure 1-3 and Figure 1-4 herein for distortional
buckling, local buckling and lateral-torsional buckling respectively.
No formal provisions for shear currently exist for Direct Strength Method.
However DSM method is currently being developed for shear capacity. The
following existing equations are recast into Direct Strength format and are
suggested for use:
forv 0.815,
100
101
102
103
0
2
4
6
8
10
12
14
16
18
20
half-wavelength
load
factor
half-wavelength (in.)
Pcr
(kips)
Local bucklingDistortional
buckling
Global
buckling
crP creP
crdP
Pcr
Pcrd
Pcre
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Vn = Vy, (1-1)
for 0.815 0.815,
Vn = Vcr, (1-3)
where,
,cryv VV
Vy = Yield shear force of web
,60.0 yw FA
Vcr= Critical elastic shear buckling force
The Finite strip method cannot predict elastic buckling in shear. Hence
simplified methods are required to predict the critical elastic buckling shear force.
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Figure 1-2: DSM distortional buckling failure design curve and equations
Figure 1-3: DSM local buckling failure design curve and equations
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
Distortional s lenderness, d=(P
y/P
crd)0.5
Pnd
/P
y
Distortional Buckling
The nominal axial strength, Pnd, for distortional buckling is
for d 561.0 Pnd = Py
for d > 0.561 Pnd = y
6.0
y
crd
6.0
y
crd PP
P
P
P25.01
where d = crdy PP
Pcrd = Critical elastic distortional column buckling load
Py = Column yield strength
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
local s lenderness, =(P
ne
/Pcr
)0.5
Pn/
Pne
Local Buckling
The nominal axial strength, Pn, for local buckling is
for 776.0 Pn = Pnefor > 0.776 Pn = ne
4.0
ne
cr
4.0
ne
cr PPP
PP15.01
where = crne PP
Pcr = Critical elastic local column buckling load
Pne = Nominal axial strength for global buckling
Local buckling interacts with
global buckling at failure
Global
failure
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Figure 1-4: DSM Global buckling failure design curve and equations
1.3 Research MotivationThe American Iron and Steel Institutes (AISI) North American
Specification (AISI-S100, 2007) calculates the shear strength of a thin walled cold-
formed steel member by treating the primary shear carrying cross-sectional
element, for example the web of a C-section, as a simply supported plate in shear
(Figure 1-5). The critical elastic buckling stress, cr, is approximated with a plate
buckling coefficient, kv:
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
Global slenderness, c=(P
y/P
cre)0.5
Pne
/Py
Flexural, Torsional, or Torsional-Flexural Buckling
The nominal axial strength, Pne, for flexural or torsional- flexural buckling is
for 5.1c Pne = yP658.02c
for c > 1.5 crey2c
ne P877.0P877.0
P
where c = crey PP
Py = AgFy
Pcre= Critical elastic global column buckling load
Ag = gross area of the column
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2
2
2
112
b
tEkvcr
(1-4)
where E is the modulus of elasticity, is the Possions ratio, b is the plate width
(Figure 1-5), and t is the plate thickness. For a cold-formed steel member with
unreinforced webs, kv=5.34, resulting from the elastic buckling solution for an
infinitely long simply-supported plate in shear (Southwell and Skan 1924). For a
reinforced web, kv is calculated with a lower bound approximation to a classical
Rayleigh-Ritz energy solution (Stein and Neff 1947; Bleich 1952; Timoshenko and
Gere 1961; Allen and Bulson 1980) assuming each reinforced web panel of depth
b and length a buckles as a simply-supported plate in shear:
234.5
00.4ba
kv
0.1ba
(1-5)
200.4
34.5ba
kv
0.1ba
The buckling stress is input into a empirically derived design expression (AISI-
S100 2007, Section C3.2.1) to calculate the ultimate strength of the member in
shear. The design approach is simple and convenient, however the beneficial
contribution provided by adjacent connected cross-section elements, for example
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the flanges of a C-section (Figure 1-5), is neglected in the shear strength
prediction.
Figure 1-5 : Web is treated as a simply-supported plate in design
Recent studies have demonstrated that cr calculated for a C-section
member including cross-section connectivity can be up to 40% higher than that
predicted by the classical solution considering just the isolated web (Pham and
Hancock 2009b). Furthermore, the critical elastic shear buckling load of a member,
Vcr, which can be calculated from cr, has been confirmed to be a viable parameter
for predicting the strength of cold-formed steel C-section members in shear and
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combined bending and shear with a Direct Strength approach (Pham and Hancock
2009a).
The AISI Direct Strength Method (DSM), introduced in 2004 and currently
limited to flexural and compression members (AISI-S100 2007, Appendix 1), has
been welcomed by the design community because of convenient freely-available
computer programs, for example CUFSM (Schafer and dny 2006), which
calculate the elastic buckling parameters for any general member cross-section.
Accessible approaches for calculating Vcrdo not currently exist however, requiring
more involved solutions employing finite element analysis or the spline finite strip
approach. Simplified methods for calculating Vcr are needed to extend the
appealing generality and accuracy of DSM to members in shear.
The goal of this research program was to develop simplified methods for
predicting the simplified methods for predicting the critical elastic buckling loads
of cold-formed steel members in shear including cross-section connectivity.
Chapter 2 focuses on the study of a Rayliegh-Ritz energy solution
employed to develop an approximate equation for classical buckling stress, cr, of
simply- supported plate in shear.
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Chapter 3 focuses on developing finite element eigen-buckling guidelines
for the commercial finite element program ABAQUS (ABAQUS 2008) and
establishing aspect ratio meshing rules for modeling thin-walled members in shear.
Chapter 4 examines the influence of cross-section connectivity on shear
buckling stress, cr. The contribution of cross-section connectivity to the buckling
stress is studied with thin shell finite element eigen-buckling analyses of cold-
formed steel members in shear, where each of the 99 structural stud C-sections
listed in the Structural Stud Manufacturers Association catalog are considered
(SSMA 2001). The results from the elastic buckling studies motivate the study of
web shear buckling including rotational restraint from connected cross-sectional
elements. Finite element eigen-buckling analysis of plates in shear with rotational
springs is performed, and a Rayleigh-Ritz energy solution for shear buckling of a
plate with rotational restraint along two edges is derived.
Chapter 5 focuses on finite element eigen-buckling analysis of the plates in
shear with rotational stiffness and in plane bending stiffness of flange and
developing an expression for a plate buckling coefficient. The computational and
analytical studies can be used to derive engineering expressions which incorporate
the beneficial effect of cross-section connectivity on shear buckling in design.
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CHAPTER 2 CLASSICAL SOLUTION FORSHEAR BUCKLING
A Rayleigh-Ritz energy solution is employed to develop an approximate
equation for the critical elastic buckling stress, cr, of a simply-supported plate in
shear (Stein and Neff 1947).The critical buckling stress is determined on the basis
of minimum potential energy. For a rectangular plate in pure shear the equation for
strain energy can be written as (Timoshenko and Gere 1961):
.12)1(122
10 0
22
2
2
2
22
2
2
2
2
2
3
dxdyyx
w
y
w
x
w
y
w
x
wEtU
a b
(2-1)
where Eis modulus of elasticity; is Poissons ratio; tis thickness of plate and a
and b are the length and width of the plate respectively. Figure 2-1 provides the
plate coordinate system and dimension notation.
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Figure 2-1: Plate coordinate system and dimension notation
The external work, W, performed on the constant shear stress, , as the
plate buckles is:
,0 0
dxdyy
w
x
wtW
a b
(2-2)
where
,2tb
Dkv
(2-3)
and kv is the shear stress coefficient, which depends upon the boundary conditions
and the aspect ratio of rectangular plate a/b. The flexural stiffness,D, of the plate
is defined as
,
112 2
3
EtD (2-4)
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where E is the Youngs modulus for material and is the Poissons ratio for the
material.
The platesbuckled shape is approximated as a double Fourier sine series:
.sinsin),(1 1
b
yn
a
xmayxw
m n
mn
(2-5)
Each term of the series in Eq. (2-5) vanishes forx = 0, x = a and y = 0, y =b,
hence the deflection is zero along the boundaries. The second derivatives 2w/x2
and 2w/y2 are also zero at the boundaries, satisfying the boundary conditions for
a simply supported plate.
For any buckle pattern where the value of w is zero at the edges, the
integral with the coefficient -2(1 - ) in Eq. (2-1) can be shown to vanish.
Integration by parts of the last term in Eq. (2-1) leads to,
dy
yx
w
x
wdx
x
w
yx
wdxdy
yx
w
yx
w
yx
w 32222
2
dxdyy
w
x
wdy
y
w
x
wdx
x
w
yx
w2
2
2
2
2
22
(2-6)
Substituting Eq. (2-5) into the first term in Eq. (2-6):
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b
n
b
yn
a
xmadx
yx
w
m n
mn
cossin
1 1
2
(2-8)
Also the second term in the integral in Eq. (2-6),
2
1 12
2
sinsin b
n
b
yn
a
xmay
w
m n
mn
b
n
b
yn
a
xmadx
y
w
m n
mn
cossin
1 12
2
(2-9)
From Eq. (2-8) and Eq. (2-9), it can be seen that the first two terms in Eq. (2-6) are
exactly identical.
Hence,
dxdyy
w
x
wdxdy
yx
w2
2
2
22
2
(2-10)
Therefore the second term in Eq. (2-1) can be shown equal to zero.
00 0
22
2
2
2
2
dxdyyx
w
y
w
x
wa b
(2-11)
The strain energy in Eq. (2-1) therefore reduces to
bn
am
byn
axma
yxw
m n
mn coscos
1 1
2
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a b
y
w
x
wEtU
0 0
2
2
2
2
2
2
3
.)1(122
1
(2-12)
Substituting Eq. (2-5) into Eq. (2-12),
.sinsin)1(122
1
0 0
2
1 12
2
2
22
2
3
dxdyb
n
a
m
b
yn
a
xma
EtU
a b m
m
n
n
mn
(2-13)
Using the trigonometric identity
(2-14)
Eq. (2-13) simplifies to:
dxdyb
n
a
ma
abEtU
m
m
n
n
mn
1 1
2
2
2
2
22
4
2
3
4)1(122
1
(2-15)
Taking the derivative of Eq. (2-5),
b
yn
a
xm
a
ma
x
w
m n
mn
sincos
1 1
.cossin1 1
b
yq
a
xp
b
qa
y
w
p q
pq
Substituting Eqs. (2-16) in Eq. (2-2):
4sinsin 2
0 0
2 abdxdyb
yn
a
xma b
(2-16)
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.cossinsincos0 0
1 1 1 1
2
dxdyb
yq
a
xp
b
yn
a
xm
ab
mqaatWa b
m n p q
pqmn
(2-17)
Observing that,
a
pm
nmapamnmamdx
a
xp
a
xm
0
22
sinsincoscoscossin
If m + p is even, then m and p must be odd. Using trigonometry,cos(m) = cos(n) = -1; and sin(m) = sin(n) = 0.
Hence,
a
dxa
yp
a
xm
0
0cossin
ifm + p is an even number.
If m + p is odd, then ifm is even and p is odd or ifm is odd thenp is
even. Again using trigonometry, cos(m) cos(n) = -1 and sin(m)sin(n) = 0.
Hence,
22
0
2cossin
pm
madx
a
yp
a
xma
ifm + p is an odd number.
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Eq. (2-17) reduces to
.
42222
1 1 1 1 qnmp
mnpqaatW
m
m
n
n
p
p
q
q
pqmn
(2-18)
where m + p and n + q are odd numbers and therefore (m + n + p+ q) must be an
even number. And ifm + n is evenp + q must also be even; ifm + n is odd thenp
+ q must also be odd.
The Rayliegh-Ritz solution is based on the principle of minimum potential
energy, i.e. (U+ W) = 0. Solving for total potential energy, i.e. (U + W= ), we
obtain the following expression:
m
m
n
n
mnb
n
a
ma
abEt
1 1
2
2
2
2
22
4
2
3
4)1(122
1
m
m
n
n
p
p
q
qpqmn qnmp
mnpq
aat 1 1 1 1 22224
(2-19)
Substituting 2
3
112
EtD
andtb
Dkv2
2 in Eq. (2-16),
m
m
n
n
mnb
n
a
ma
abD
1 1
2
2
2
2
22
4
42
(2-20)
.
4
1 1 1 1 22222
2
m
m
n
n
p
p
q
q pqmn
v
qnmp
mnpqaa
b
Dk
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It is necessary to select a system of constants amn and apq so as to make kv
minimum, i.e. (U + W) = 0. Taking the derivative of Eq. (2-20) with respect to
each of the coefficients amn, i.e. /amn = 0, we obtain a system of equation in
terms ofamn, represented by the following:
.0
32
1 1
22222
3
22
22
p
p
q
q
pq
v
mn
qnpm
mnpqa
b
ak
b
anma
(2-21)
Eq. (2-21) can be further simplified as
,0
321 1
2222
22
22
3
2
p
p
q
q
pqmn
v
qnpm
mnpqaa
b
anm
b
ak
(2-22)
or
01 1
p
p
q
q
pqnqmpmnmn aCCaL (2-23)
22
22
3
2
32
b
anm
b
ak
L
v
mn
,
22 pmmp
Cmp
, .
22 qn
nqCnq
Eq. (2-23) represents a system of linear equations in terms of the Fourier
coefficients, amn and apq. This system can be divided into two groups which are
independent of each other, one containing constants amn in which m + n is even
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(symmetric buckling modes) and the other for which m + n is odd (anti-symmetric
buckling modes) as shown in Figure 2-2.
Figure 2-2 : (a) Symmetric buckle mode (b) anti-symmetric buckle mode
An exact solution for critical shear stress for a rectangular plate involves
the use of an infinite set of equations in an infinite number of unknowns. Since
attention must be confined to a finite number of equations, say N, the ability to
choose the bestNequations for the purpose is very desirable.
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A finite number of Fourier series terms are assumed and Eq. (2-23) is
solved simultaneously for the buckling coefficient, kv (Stein and Neff 1947). The
buckled shape shown in Figure 2-2a has amn terms where m + n even. As shown in
Table 2-1 form=1 and n=1, the values ofp and q range from 1 to 7, so that the
number of Fourier terms apq is set to 20.
A very useful guide to the best choice of equations to be used may be
obtained from a consideration of the accuracy of representation of the buckling
deformation. The use ofNnumber of equations implies that the deflection surface
is being described in terms ofNFourier components, with the other components
are equal to zero. The values found for the Fourier coefficients whereNwas taken
as 20 are substituted in the form as shown in Figure 2-3. As a result of this
substitution, values are inserted in the 20 squares corresponding to the coefficients
assumed not equal to zero, whereas no values were substituted for the remaining
squares.
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Figure 2-3: Choice of Equations (a) m +n even (b) m +n odd
The equation chosen for each particular value of a/b should contain
deflection coefficients that give the lowest values kv for each type of buckling.
Fourier coefficients for any general a/b are given in the Table 2-1 .
A system of linear equations can be written using Eq. (2-23) as follows:
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.0.........3111132212121313111111 pqnqmp aCCaCCaCCaCCaL.0.........
3121232232121313113111 pqnqmp aCCaCCaCCaLaCC
.0.........3121232222132321112121
pqnqmp aCCaCCaLaCCaCC (2-24)
.....................................................................................................
.0.........3113222213311111 mnmnnmnmnmnm aLaCCaCCaCCaCC
Eqs. (2-24) can be written in the matrix form as,
or concisely,
[A] [X] = [0]. (2-26)
L 11 C11 C13 C12 C12 C13 C11 ... Cmp Cnq
C11 C31 L 13 C12 C32 C13 C31 ... Cmp Cnq
C21 C21 C21 C23 L 22 C23 C2 1 ... Cmp Cnq
... ... ... ... ... ...
Cm1 Cn1 Cm1 Cn3 Cm2 Cn2 Cm3 Cn1 .. L mn
a 11
a 13
a 22
.
a mn
0
0
= 0
.
0
(2-25)
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Table 2-1: Representative determinant in terms of coefficients for the group of equations in which (m +n) is even
.
32
22
22
3
2
b
anm
b
ak
L
v
mn
m,na11 a13 a22 a31 a15 a24 a33 a42 a51 a17 a26 a35 a44 a53 a62 a71 a37 a46 a55 a64
m=1, n=1 L11 0 4/9 0 0 8/45 0 8/45 0 0 4/35 0 16/225 0 4/35 0 0 8/175 0 8/175
m=1, n=3 0 L13 -4/5 0 0 -8/7 0 -8/25 0 0 4/9 0 16/35 0 -36/175 0 0 8/45 0 72/245
m=2, n=2 4/9 -4/5 L22 -4/5 -20/63 0 36/25 0 -20/63 -28/135 0 4/7 0 4/7 0 -28/135 28/75 0 100/441 0
m=3, n=1 0 0 -4/5 L31 0 -8/25 0 8/7 0 0 -36/175 0 16/35 0 4/9 0 0 72/245 0 8/45
m=1, n=5 0 0 -20/63 0 L15 -40/27 0 -8/63 0 0 20/11 0 -16/27 0 -40/49 0 0 8/11 0 -8/21
m=2, n=4 8/45 -8/7 0 -40/27 L24 -72/35 0 -8/63 -56/99 0 8/3 0 -40/49 0 -56/675 56/55 0 200/11 0
m=3, n=3 0 0 36/25 0 0 -72/35 L33 -72/35 0 0 -4/5 0 144/49 0 -4/5 0 0 9/7 0 8/7
m=4, n=2 8/45 -8/25 0 8/7 -8/63 0 -72/35 L42 -40/27 -56/675 0 -40/49 -120/47 8/3 0 -56/99 -8/15 0 200/11 0
m=5, n=1 0 0 -20/63 0 0 -8/63 0 -40/27 L51 0 -4/49 0 -16/27 0 20/11 0 0 -8/21 0 8/11
m=1, n=7 0 0 -28/135 0 0 -56/99 0 -56/675 0 L17 -28/13 0 -112/495 0 -4/75 0 0 -56/65 0 -8/55
m=2, n=6 4/35 4/9 0 -36/175 20/11 0 -4/5 0 -4/49 -28/13 L26 36/11 0 -20/63 0 -4/75 252/65 0 -100/77 0
m=3, n=5 0 0 4/7 0 0 8/3 0 -40/49 0 0 -36/11 L35 -80/21 0 -20/63 0 0 360/77 0 -40/27
m=4, n=4 16/225 16/35 0 16/35 -16/27 0 144/49 -120/47 - 16/27 -112/495 0 -80/21 L44 -80/21 0 -112/495 -16/11 0 400/81 0
m=5, n=3 0 0 4/7 0 0 -40/49 0 8/3 0 0 -20/63 0 -80/21 L53 -36/11 0 0 -40/27 0 360/77
m=6, n=2 4/35 -36/175 0 4/9 -40/49 0 -4/5 0 20/11 -4/75 0 -20/63 0 -36/11 L62 -28/13 -28/135 0 -100/77 0
m=7, n=1 0 0 -28/135 0 0 -56/675 0 -56/99 0 0 -4/75 0 -112/495 0 -28/13 L71 0 -8/55 0 -56/65
m=3, n=7 0 0 28/75 0 0 56/55 0 -8/15 0 0 252/65 0 -16/11 0 -28/135 0 L37 -72/13 0 -56/99
m=4, n=6 8/175 8/45 0 72/245 8/11 0 9/7 0 -8/21 -56/65 0 360/77 0 -40/27 0 -8/55 -72/13 L46 -200/33 0
m=5, n=5 0 0 100/441 0 0 200/11 0 200/11 0 0 -100/77 0 400/81 0 -100/77 0 0 -200/33 L55 -200/33
m=6, n=4 8/175 72/245 0 8/45 -8/21 0 8/7 0 8/11 -8/55 0 -40/27 0 360/77 0 -56/65 -56/99 0 -200/33 L64
apq
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The equation for calculating kv, are obtained by selecting a numerical
value for the length-width ratio a/bandby equating to zero the determinant of the
above system of equations and solving for the lowest value of kv that satisfies
determinant [A] = 0. Table 2-1provides coefficients for the group of equations in
which m + n is even. A case should also be considered in which m + n is odd. The
lower of the two values ofkv found from the two determinants will produce the
critical elastic buckling stress for a plate with length-width ratio of a/b. The
values of the deflection function coefficients amn and apq can also be obtained
using Eq.(2-24). The accuracy of this solution increases with more series terms. A
solution for cr over a plate aspect ratio, a/b is provided with 10 simultaneous
equations in Stein and Neff (1947), however in this chapter, 20 equations with 20
unknowns were solved to ensure a viable comparison of following finite element
studies. Figure 2-4 shows a comparison between the solutions obtained by using
10 and 20 Fourier series terms.
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Figure 2-4 : Comparison between solutions using 10 and 20 Fourier series terms.
As the number of terms is increased, the plate buckled shape is
approximated more accurately. Hence the accuracy of the solution increases as the
number of terms in the series is increased. Figure 2-4 also shows a comparison of
the classical solution with the commonly used approximation for kv (Eq. 1-2) as
described in Chapter 1.
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The theoretical solution described above is used in the following chapter
to develop and validate a finite element modeling protocol for buckling of thin
plates in shear.
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CHAPTER 3 FINITE ELEMENT MODELINGGUIDELINES FOR THIN
PLATES IN SHEAR
Finite element modeling guidelines are established in this chapter for
eigen-buckling analysis of thin plates in shear. Finite element analysis is an
effective tool to study buckling of thin-walled structures. Accuracy of the analysis
depends on several factors including the type of the element, the meshing
geometry and density and the assumed boundary conditions. Parameter studies are
carried out to compare finite element eigen-buckling predictions to the theoretical
solutions presented in Chapter 2, quantify the accuracy of ABAQUS thin shell
elements and to identify limits on element aspect ratio and element density.
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3.1 Summary of ABAQUS thin-shell elementsThree ABAQUS finite elements commonly employed in the elastic
buckling analysis of thin-walled structures are the S4, S4R, and S9R5 elements as
shown in Figure 3-1.
Figure 3-1: a) S4/ S4R shell element (b) S9R5 shell element
The S4 and S4R are four node general purpose shell elements valid forboth thick and thin shell problems (ABAQUS 2008). Both elements use linear
shape functions to interpolate displacements between nodes. The S4 element
employs a normal integration rule with four integration points as shown in Figure
3-2a. The S4R element uses a reduced integration rule with one integration point
as shown in Figure 3-2b that makes this element computationally less expensive
than the S4 element. Reduced integration also helps avoid shear locking.
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Figure 3-2: a) Four point integration rule (b) One point integration rule
The ABAQUS S4 element uses a theory similar to Mindlin theory (Cook
1989) in its formulation. In this theory the transverse shear deformation is
included by relaxing the assumption that plane sections remain perpendicular to
middle surface, i.e. right angles in the element are no longer preserved. A plate of
thickness t has a midsurface at a distance t/2 from each lateral surface. For
analysis, we locate the xy plane in the plate midsurface (Figure 3-3) so that z = 0
identifies the middle surface.
Figure 3-3 : A plate element with corner nodes showing a midsurface.
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The S4R element uses a mixed finite element formulation. In this
formulation, neither pure Kirchoff nor pure Mindlin plate theory is used.
Kirchoffs theory (Cook 1989) ignores the effects of transverse shear
deformation. Due to the reduced number of integration points, hourglassing can
occur in the S4R element. An hourglass stabilization control feature is built into
the element; therefore, ABAQUS automatically checks for the possible hourglass
mode shapes.
The S9R5 element is a doubly-curved thin shell element with nine nodes
derived with shear flexible Mindlin strain definitions and Kirchoff constraints
(classical plate theory with no transverse shear deformation) enforced as penalty
functions (Schafer 1997). This element uses quadratic shape functions to
interpolate displacements between nodes (resulting from the increase in number
of nodes from 4 to 9). The quadratic shape function provides the ability to define
initially curved geometries and approximates a half sine wave with just one
element. The 5 in S9R5 denotes that each element node has 5 degrees of
freedom (three translational, two rotational) instead of 6 (three translational, three
rotational). The rotation of a node about the axis normal to the element
midsurface, i.e., the rotation about z-axis, is removed from the element
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formulation to improve computational efficiency. The S9R5 element also uses
reduced integration. The accuracy of eigen-buckling finite element models are
compared here for each of these ABAQUS S4, S4R and S9R5 elements against
the exact solutions.
3.2 Loading and Boundary conditions
The critical elastic buckling stress, cr, for a thin plate in pure shear is
determined by applying the unit stress distribution as shown in Figure 3-4 and
then performing an eigen-buckling analysis. The shear stress is simulated in
ABAQUS by applying a consistent nodal load, Vnode, at each node:
a
noden
taV or
b
nodentaV (3-1)
where is a unit shear stress (i.e. = 1 ksi) and na and nb are the number of nodes
along the length or width of the plate respectively.
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Figure 3-4: Plate boundary conditions and loading
Note that a rectangular plate loaded with a constant shear stress is not in moment
equilibrium about the 3-axis (Figure 3-4) and therefore special attention is
required when applying the boundary conditions to prevent rigid body rotation in
the finite element model. The classical formulation described in Chapter 2
considers only out-of-plane deformations, w, and therefore the in-plane force
imbalance does not affect the solution. To address the moment imbalance in
ABAQUS, reactions are provided at opposing corners of the plate (Figure 3-5).
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Figure 3-5 : Rigid body rotation restrained in ABAQUS
3.3 Summary of ABAQUS thin-shell elementsFinite element elastic buckling analyses of thin rectangular plates in shear
were performed to compare ABAQUS predictions to the classical energy solution.
The plate thickness is t = 0.0346 in., E = 29500 ksi and = 0.3 for all finite
element models considered in the study. The plate dimensions were held constant
at a=20 in. and b=5 in., with 3 buckled half-waves forming along the length of the
plate (refer Figure 2-1 fora and b definitions).
For plate buckling problems, it is convenient to think of mesh density as a
function of buckled half-wavelength along the plate. In other words, how many
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elements are required per half-wave to accurately represent the true buckled
shape?
The quantity of elements along the plate width, b, are fixed to 20 elements
and the quantity of elements along the length, a, is varied gradually, thereby
varying the number of elements required to model a single buckled half-wave.
The plate models implemented with the S9R5 element converge to the classical
energy solution in Figure 3-6 as the number of elements per half-wave increase.
The S4 and S4R element solutions converge to a constant buckling stress that is
5% higher than the classical solution. The S9R5 element is within 1% of the
classical solution for 2 elements per half-wave while 6 elements per half-wave are
required to achieve a similar accuracy for the S4 and S4R element. The S4
element experiences membrane locking when the number of elements per half-
wave is less than six elements, resulting in a buckling stress up to 60% higher
than the theoretical value. The S4R avoids this membrane locking with a reduced
integration scheme that assumes the membrane stiffness is constant in the
element. However, the results degrade when less than four elements per half-
wavelength are used.
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Figure 3-6: Variation in ABAQUS predicted buckling coefficient kv with number of elements
per buckled half-wave
Also, as can be seen from the buckled shape, diagonal buckling occurs in
the members. The S4 and S4R elements use linear shape functions to estimate
displacements thereby requiring more number of elements to capture the buckled
shape. On the other hand, the S9R5 element uses a quadratic shape function to
estimate displacements and can therefore capture the buckled shape of the plate
with less number of elements.
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The same rectangular plate (a=20 in., b=5 in.) is now employed in a
separate study evaluating the influence of element aspect ratio, A/B, on solution
accuracy. The number of elements per half-wave is fixed at 10 along the plate
length, a, as it was observed that kv converges to a constant magnitude for all
element types in Figure 3-6 with 10 number of elements.
Figure 3-7: Variation in ABAQUS predicted buckling coefficient kv with element aspect
ratio, A/B
The element aspect ratio is then varied by changing the number of
elements along the plate width, b. The finite element plate model with S9R5
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elements is consistent with the classical solution for element aspect ratios between
0.5 to 4.0 as shown in Figure 3-7, while the S4 and S4R elements performs most
accurately (within 10% of classical solution) for element aspect ratios between 1.0
and 2.0.
In summary, the following ABAQUS meshing and aspect ratio guidelines are
recommended when performing eigen-buckling analysis of thin-walled members
in shear:
S9R5 element - use 2 or more elements per half-wave with an elementaspect ratio between 0.5 and 4.
S4R and S4 element - use 6 or more elements per half-wave with anelement aspect ratio between 1 and 2.
Using the newly introduced meshing guidelines, models are created to
compare eigen-buckling solution accuracy of the S4, S4R, and S9R5 elements for
varying plate aspect ratios as shown in Figure 3-8. Node for node, the S9R5
element is more accurate than the S4 and S4R element, which is consistent with
similar elastic buckling studies of uniaxially compressed simply-supported plates
(Moen 2008).
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Figure 3-8 : Variation in ABAQUS predicted buckling coefficient kv with plate aspect ratio,
a/b
The buckling coefficient, kv, remains within 1% of the classical solution
across the range of plate aspect ratios (between 1 and 4) considered when
modeling with the S9R5 element. The accuracy of the plate models with S4 and
S4R elements increase with increasing plate aspect ratio. The S4 element is
observed to be slightly stiffer than the S4R and S9R5 element which is
hypothesized to occur as a result of the full integration stiffness calculation.
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CHAPTER 4 SHEAR ELASTIC BUCKLING STUDIESON CHANNEL SECTIONS
The finite element model guidelines developed in the previous chapter are
utilized to calculate crof lipped C-section cold-formed steel members. The goal
of this study is to quantify the increase in cr provided by cross-section
connectivity in industry standard structural stud cross-sections. A finite element
eigen-buckling analysis was conducted on each of the 99 C-section structural stud
cross-sections listed in the Structural Stud Manufacturers Association (SSMA)
catalog (SSMA2001) for comparing the critical buckling stress, cr, for the full
member to the crfor the plate.
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4.1 Finite Element Modeling AssumptionsThe elastic buckling behavior of the coldformed steel structural members
are evaluated in this chapter using thin shell finite element eigen buckling
analyses in ABAQUS (ABAQUS 2008). All members are modeled with
ABAQUS S9R5 reduced integration ninenode thin shell elements. The typical
finite element aspect ratio is 1:1 and the maximum aspect ratio is limited to 4:1
(refer to Chapter 3 for a discussion on ABAQUS thin shell finite element types
and finite element aspect ratio limits). Element meshing is performed with custom
Matlab (Mathworks 2009) code. Coldformed steel material properties are
assumed as E=29500 ksi and =0.3 in the finite element models. The length of
each member was established by maintaining the member length L=4H in first
case and a member length ofL=8H in second case, where H is the out-to-out
depth of the cross-section (Figure 4-1).
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4.2 Loading and Boundary ConditionsThe ABAQUS member boundary conditions and loading are described in
Figure 4-1 and are the same as those employed for the plate studies (Figure 3-4).
The web of each member is loaded with a unit shear stress.
Figure 4-1: (a) Member loading, boundary conditions and dimension notation, (b) cross-
section dimension range where H, B, L, D are out-to-out dimensions, r is the
inside radius and t is the thickness.
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4.3 Elastic Buckling AnalysesThis study builds on the results and observations in Chapter 3 for thin
plates in shear and marks a transition in research focus from plate elements to full
cold-formed steel members. The influence of cross-section connectivity on shear
buckling stress of SSMA coldformed steel structural sections is summarized, the
goal being the development of simplified method for predicting the critical elasticbuckling load of cold-formed steel members in shear including cross-section
connectivity. The plate widths are chosen to correspond with the flat web widths
of standard SSMA structural studs (SSMA 2001). For each plate a full structural
finite element model is developed for comparison. The range of cross-section
dimensions considered is summarized in Figure 4-1b.
Before examining the buckling stress, consider the observed changes in
the first mode shape caused by the addition of the cross-section. For the buckled
shape of the SSMA 1200S20054 in Figure 4-2a, the number of buckled
halfwaves changes from 6 for the isolated plate to 8 for the full member. A
similar trend is observed when SSMA 800S200-54 (Figure 4-2b) and SSMA
400S200-54 (Figure 4-2 c) is considered.
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Figure 4-2: (a) SSMA 1200S200-54 web plate and structural stud, (b) SSMA 800S200-54 web plate and structural
stud, and (c) SSMA 400S200-54 web plate and structural stud.
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The results of the buckling analyses of channel sections for an aspect ratio
(L/H) of 8.0 are shown in Figure 4-3. The relationship between the ratio of flange
and web width (H/B) and the shear buckling coefficient is plotted. Figure 4-3
illustrates that crincreases by as much as 50% when cross-section connectivity is
considered which is consistent with other recent research (Pham and Hancock
2008).
Figure 4-3: Variation in crwith H/B for SSMA sections for L/H=8.0.
The increase in buckling stress occurs because of the rotational restraint
provided to the web by the flange and stiffening lip. The increase in cris largest
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when H/B is small. The observed beneficial contribution of cross-section
connectivity is explored further with finite element modeling and the classical
energy solution for shear buckling in the next section.
4.4 Extension of the classical shear buckling equation for plates withrotational springs
Considering cross-section connectivity has the potential to improve shear
buckling capacity of thin walled sections. An effort has been made in this section
to study elastic buckling of plates supported by rotational springs along the
longitudinal edges. The Rayleigh-Ritz energy solution presented in Chapter 2 has
been modified to include the rotational springs (Figure 4-4) simulating this cross-
section connectivity.
A spring stiffness, kr, in units of (forcelength)/radian per unit length is
used to simulate the rotational restraint provided by flanges connected to the web.
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Figure 4-4: Plate coordinate system and dimension notation
The critical buckling stress is based on the principle of minimum potential energy,
i.e. (U+ W) = 0. For a rectangular plate in pure shear the equation for strain
energy in Eq. (2-1) can be written with an additional strain energy term
accounting for a spring, kr, distributed along the plate edges:
dxdyyx
w
x
w
x
w
y
w
x
wEtU
a b
0 0
22
2
2
2
22
2
2
2
2
2
3
12)1(122
1
a
r dxxk0
2,
where rotation at the longitudinal plate edges is defined as
(4-1)
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.0,xyxwx
(4-2)
The external work, W, performed on the constant shear stress, , is assumed to be
unchanged with the addition of the springs and is given as:
.0 0
dxdyy
w
x
wtW
a b
(4-3)
The boundary conditions at the supported edges are satisfied by taking thedeflection surface of the buckled shape in the form of double Fourier series as
.sinsin),(1 1
b
yn
a
xmayxw
m n
mn
(4-4)
Substituting Eq.(4-4) into Eq.(4-1),
2
1 12
2
2
22
2
3
sinsin)1(122
1
m
m
n
n
mnb
n
a
m
b
yn
a
xma
EtU
xdxb
yn
a
xm
b
nak mn
a
r
2
0
sinsin
(4-5)
Then observing that
,4sinsin0
22
0
a b
r
ab
dxdyb
yn
a
xm
k
(4-6)
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we obtain,
m
m
n
n
mnb
n
a
ma
abEtU
1 1
2
2
2
2
22
4
2
3
4)1(122
1
(4-7)
.21 1
22
2
m n
mnr
n
b
aak
The work done remains unchanged as in Eq. 3-2, given as:
.
42222
1 1 1 1 qnmp
mnpqaatW
m
m
n
n
p
p
q
q
pqmn
(4-8)
Solving for total potential energy, i.e. (U + W = ), we obtain the following
expression:
1 1
22
2
1 1
2
2
2
2
22
4
2
3
24)1(122
1
m n
mnr
m
m
n
n
mn
n
b
aak
b
n
a
ma
abEt
(4-9)
22221 1 1 14
qnpm
mnpqaat
m
m
n
n
p
p
q
q
pqmn
It is necessary to select a system of constants amn and apq so as to make kv
minimum, i.e the total potential energy is zero ((U + W)=0). Taking the
derivative of the Eq. (4-9) with respect to each of the coefficients amn, i.e.
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/amn= 0, we obtain a system of equation in terms ofamn, taking into account
the effect of rotational springs and represented by the following:
.01384148
1 1222224
2323
3
22
22
p
p
q
q
mnrmnqnpm
mnpq
bEt
a
n
m
b
a
Et
aak
b
anma
(4-10)
The critical elastic buckling stress, cr, including the restraint provided by
rotational springs is obtained by selecting a finite number of Fourier series terms
and solving Eq. (4-10) simultaneously for buckling coefficient, kv. This system
can be divided into two groups which are independent of each other, one
containing constants amn in which m + n is even (symmetric buckling modes) and
the other for which m + n is odd (anti-symmetric buckling modes). The Eq. (4-10)
was further solved as explained in Chapter 2.
4.5 Verification by Finite-Element AnalysisFinite element analysis using ABAQUS was employed to examine the
analytical results for a plate with rotational spring along the longitudinal edges.
The plate properties are Youngs modulusE= 29500 ksi and Poissons ratio =
0.3 and thickness t = 0.0346 in. The finite element loading and boundary
conditions are the same as shown in Figure 3-4 in Chapter 3 with the addition of
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rotational springs modeled as SPRING1 elements. SPRING1 element had 6
degrees of freedom (3 translational and 3 rotational). This type of element has one
end attached to the ground, i.e., a rigid surface, and the other end attached to a
node in the model as shown in Figure 4-5. The spring provides stiffness to a
global degree of freedom. The direction of action for SPRING1 elements are
defined by giving the degree of freedom at each node of the element. This degree
of freedom may be in a local coordinate system.
Figure 4-5: Spring element between node and the ground with 6 degrees of freedom
(SPRING1)
ABAQUS shell element S9R5 was used to model the plate elements. The
plate width is kept constant at 8 in. and the length is varied from 8 in. to 32 in
Figure 4-6 compares the buckling stress, cr, derived in Eq. (4-10) to finite
element eigen-buckling solutions over a range of plate aspect ratio (a/b) with the
rotational stiffness of the spring as kr= 0.5 kip.in./radian per in.
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Figure 4-6: Comparison ofkv calculated with energy solution to FE eigen buckling solutionwith rotational restraint
The energy approximation trends with the finite element solution,
predicting values ofkv that deviate above the FE solution by as much as 15 % as
(a/b) increases. The energy solution accuracy could be improved by using more
Fourier sine series terms or by considering a different function for w(x) that can
more accurately simulate the slope of the plate, (x) in Eq. (4-2), at the restrained
edges. It should also be noted that for a/b > 4, kv, unexpectedly trends upward
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when solved with the energy solution in Eq. (4-10), even for the case without
rotational restraint, i.e. kr=0. This result is inconsistent with finite element results
beyond a/b = 4 and the classical solution for an infinitely long plate where kv
converges to a constant magnitude.
The finite element solution shown in Figure 4-6 is expanded to other
values ofkrin Figure 4-7. The values ofkrare varied from 0 to 4.0 kip.in./radian
per in. The buckling stress, cr, increases when rotational restraint is provided to
the plate edges, which is consistent with the trend for full cold-formed steel
members in shear presented in Figure 4-3.
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Figure 4-7: Variation in kv with plate aspect ratio a/b and k
The results obtained in Figure 4-7 can be used to predict the critical
buckling stress, cr, of the cold-formed steel members with cross-section
connectivity if convenient procedures for prediction kv are developed.
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4.6 Design Expression for Plate BucklingThe analyses demonstrated that cross-section connectivity has significant
impact on the elastic shear buckling stress of the members. Finite element
analysis is employed to calculate the elastic buckling stress of the cross-section
for varying aspect ratio and rotational stiffness. The plot for variation in buckling
coefficient with variation in the rotational stiffness of the springs is plotted in
Figure 4-7. The goal is to determine a simplified equation for plate buckling
coefficient which takes into account the effect of rotational stiffness of the
springs. The expression for buckling coefficient of a plate without spring stiffness
is given as:
2
434.5
b
akv (4-11)
The above equation forms the basis for the expression for the plate buckling
coefficient of plate including the rotational stiffness of the plate.
Curve fitting in the Figure 4-7 results in the equation:
06.06.2
24.15.0
434.5
k
v bak
ba
k (4-12)
where (a/b) = plate aspect ratio, k= rotational stiffness of the spring.
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Figure 4-8: Family of curves used to simulate the effect of rotational springs on the buckling
capacity.
Figure 4-8 shows the comparison between ABAQUS solution and the design
expression in Eq. (4-12). The average percentage error between the two solutions
is 3.5%, with a maximum error of 3.6%. The coefficient of determination, R2, is
0.9963 and the root mean square error is 0.04314.
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The chapter just presented focus on developing a Rayliegh-Ritz solution
for plates with rotational restraint along its longitudinal edges. The next chapter
uses results from this energy solution to quantify the influence of cross-section
connectivity of shear buckling stress.
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CHAPTER 5 DESIGN IMPLEMENTATION
The cross-section connectivity between the flange and the web can be
defined by using a rotational stiffness along with an in plane bending stiffness.
The buckled shape of an SSMA 800S200-54 section is shown in Figure 5-1. As
can be seen in the figure, the diagonal buckling is seen in the web of the cross-
section. In order for this diagonal buckling to occur, there is an in plane bending
stiffness component in addition to the rotational stiffness component. An attempt
has been made in this chapter to analyze the combined effect of in plane bending
stiffness and rotational stiffness on the plate buckling stress and to compare it
with the solution obtained from ABAQUS.
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Figure 5-1 : Buckled shape of an SSMA 800S200-54 section
5.1 Rotational stiffness
A plate model is developed in ABAQUS to study the rotational stiffness
provided by the flange to the web of SSMA 800S200-54 section. The plate
dimensions in ABAQUS are chosen to correspond to the flange of the 800S200-
54 section over one distortional half-wave. The plate width h is 2.0 in., the plate
lengthL=8 inches , and t=0.0566 in. The modulus of elasticity, E, is assumed as
29500 ksi and Poissons ratio, , as 0.3 for all finite element models considered
here. The ABAQUS boundary conditions and applied loading are described in
Figure 5-2. The plate is simply supported and loaded with imposed rotations at the
long edges of the plate with magnitudes varying as a half-sine wave to simulate
deformation over one half wavelength.
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At each node where an imposed rotation is applied, the associated moment
is obtained from ABAQUS and plotted in Figure 5-3 as a rotational stiffness per
unit length.
Figure 5-2: ABAQUS boundary conditions and imposed rotation for the web plate
Figure 5-3: Rotational stiffness of the plate
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5.2 In plane bending stiffnessThe plate model similar to the one used for computation of rotational
stiffness is now developed for the study of in plane bending stiffness provided by
the flange to the web of the SSMA 800S200-54 section. The ABAQUS boundary
conditions and applied loading are described in Figure 5-4. The plate is simply
supported and loaded with imposed rotations at the long edges of the plate withmagnitudes varying as a half-sine wave to simulate deformation over one half
wavelength.
Figure 5-4: ABAQUS boundary conditions and imposed rotation for the web plate
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At each node where an imposed rotation is applied, the associated moment
is obtained from ABAQUS and plotted in Figure 5-5 as an in plane bending
stiffness per unit length.
Figure 5-5: In plane bending stiffness of the plate
5.3 Verification by Finite-Element AnalysisFinite element models are developed to compare the buckling capacity of
the plates with rotational and in plane bending stiffness to the buckling capacity
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of the entire cross-section. The first step is to consider an entire SSMA cross-
section for analysis. This cross-section would be analyzed for different aspect
ratios. The next step is to model a plate simulating the web of the SSMA section.
This plate would be modeled with both rotational and in plane bending stiffness to
consider the stiffness provided by the flanges to the web of the cross-section.
An SSMA 800S200-54 section is considered for analysis. The modulus of
elasticity, E, is assumed as 29500 ksi and Poissons ratio, , as 0.3 for all finite
element models considered here. The cross-section aspect ratio is varied from 1.0
to 4.0. The loading and the boundary conditions are the same as that shown in
Figure 4-1a in chapter 4.
A plate model is developed in ABAQUS to study the effect of rotational
stiffness and in plane bending stiffness on the buckling capacity. The plate
dimensions in ABAQUS are chosen to correspond to the web of the SSMA
800S200-54 section. The plate aspect ratio is also varied from 1.0 to 4.0. Spring
stiffness kr and kt are used simulate the rotational and the in plane bending
stiffness provided by the flanges to the web. The loading conditions for the plate
are shown in Figure 5-6. The boundary conditions are similar to the plate in
Figure 3-4 in chapter 3. The magnitude of the rotational and the in plane bending
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stiffness is obtained from Figure 5-3 and Figure 5-5 respectively. The rotational
and in plane bending stiffness are modeled using ABAQUS SPRING1 element
(Figure 4-5).
Figure 5-6: ABAQUS loading conditions for the plate
Figure 5-6 shows the comparision of the buckling stress, cr, for a C-
section and a simply supported plate with rotational stiffness of 0.67 kips-
in./rad/in. and in plane bending stiffness of 5.41 kips-in./rad/in. The plot also
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compares the buckling stresses of plates with no springs and plate with a
rotational spring stiffness of 0.67 kips-in./rad/in.
Figure 5-7: Variation in kv with plate aspect ratio and kr and kt
As can be seen from the plot, when the flange is simulated by only a
rotational stiffness of 0.67 kips-in./rad/in. the buckling stress for a plate is within
10% of the C-section. However when a in plane bending stiffness of 5.41 kips-
in./rad/in. is added to the rotational stiffness of 0.67 kips-in./rad/in. the buckling
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stress is within 1% of that of the C-section. This can be attributed to the fact that
when an in plane bending stiffness is used along with a rotational stiffness, it
accurately approximates the stiffness provided by the flange to the web.
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CHAPTER 6 CONCLUSIONS AND FUTURE WORK
6.1 Conclusions
The elastic buckling of thin plates were studied with shell finite element
eigen-buckling analysis. A classical energy solution for thin walled members
proposed by Dr. Manuel Stein was studied in detail. Both these studies resulted in
finite-element meshing guidelines for thin-walled members in shear. These
guidelines can be summarized as follows:
While using the ABAQUS S9R5 element for eigen-buckling analysis ofthin-walled members use 2 or more elements per half wave with an
element aspect ratio between 0.5 and 4.
While using the ABAQUS S4 or S4R element for eigen-buckling analysisof thin-walled members use 6 or more elements per half wave with an
element aspect ratio between 1 and 2.
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The plate studies also confirmed that node for node the, ABAQUS S9R5
element is a more versatile performer and gives accurate results than the S4R and
the S4 elements when modeling thin walled members in shear. The S4 element is
S4R and S9R5 element which is also consistent with the previous research. Also
the number of elements per half-wavelength (physical scale) and element aspect
ratio (relative scale) are useful quantities for defining general FE mesh density
guidelines.
Finite element models of C-section members in shear were developed which
demonstrated that cross-section connectivity can increase the critical elastic
buckling stress by as much as 50% when compared to the traditional assumptions
of a simply-supported plate.
Motivated by this observation, finite element eigen buckling solutions and
Rayliegh-Ritz shear buckling energy approximate, both including rotational
springs, were developed to quantify cross-section connectivity on shear buckling
stress. A simplified equation for the buckling coefficient of the plate ,kv, including
rotational restraint is proposed and can be stated as:
06.06.2
2 4.15.04
34.5
k
v bak
bak
(6-1)
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where (a/b) is the plate aspect ratio and kris the rotational stiffness of the spring.
Further studies showed that in addition to the rotational stiffness there is
also a in plane bending stiffness provided by the flange to the web. Finite element
models were developed to analyze the combined effect of rotational and in plane
bending stiffness on the shear buckling stress. The study showed that a combined
use of rotational stiffness and in plane bending stiffness gives results that
converge with the results for entire channel sections.
.
6.2 Recommendations for future research
From the present research an expression to predict the buckling coefficient
of the plate ,kv, including rotational restriant . It is proposed to find expression to
predict buckling coefficient of plate when both rotational and in plane bending
restraint is considered. These expressions for buckling coefficient can be used to
predict the critical elastic buckling stress of cold-formed steel members including
cross-section connectivity if convenient procedures for predicting kr and kt can
be developed. It is recommended to develop hand calculations forkr and ktformembers in shear.
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REFERENCES
ABAQUS. (2009a). ABAQUS/Standard Version 6.7-1. Dassault Systmes,
http://www.simulia.com/Providence, RI.
AISI. (1996). Cold-Formed Steel Design Manual, Washington. D.C.
AISI-S100. (2007). North American Specification for the Design of Cold-Formed Steel Structural
Members, American Iron and Steel Institute, Washington, D.C.
Allen, H.G., and Bulson, P.S. (1980). Background to Buckling, McGraw-Hill, Maidenhead.
Batdorf, S.B., and Stein, M. (1947). "Technical Note No. 1223: Critical combinations of shear and
direct stress for simply supported rectangular flat plates." NACA,Langley Field, VA.
Bliech. F. (1952). Buckling Strength of Metal Structures, McGraw-Hill, New York, NY.
Cheung, Y.K., and Tham, L.G. (1998). Finite strip method. Boca Raton, Florida, FL.
Cook, R.D. (1989). Concepts and Applications of Finite Element Analysis, J. Wiley &Sons, NewYork, NY.
Hancock, G. J. (2001). Cold-Formed Steel Structures to the AISI Specification. New York, Marcel
Dekker, Inc.
Kreyszig, E. (1993). Advanced Engineering Mathematics. J. Wiley &Sons, New York, NY.
Logan, D.L. (1993). A first course in finite element method, PWS Pub.Co. Boston.
Mathworks. (2009). "Matlab 7.5.0 (R2009a)." Mathworks, Inc.,www.mathworks.com.
http://www.simulia.com/Providencehttp://www.simulia.com/Providencehttp://www.mathworks.com/http://www.mathworks.com/http://www.mathworks.com/http://www.mathworks.com/http://www.simulia.com/Providence7/28/2019 Naik Shearbuckling
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Moen, C. D. (2008). Direct Strength Design of Cold-Formed Steel Members with
Perforations.Department of Civil Engineering Baltimore, Johns Hopkins University.
Doctor of Philosophy: 592.
Moen, C. D. and B. W. Schafer (2009). "Elastic buckling of thin plates with holes in compression
or bending." Thin-Walled Structures 47(12): 1597-1607.
Pham, C.H., and Hancock, G.J. (2009a). "Experimental investigation of high strength cold-formed
c-section in combined bending and shear." Research Report - University of Sydney,
Department of Civil Engineering(894), 1-42.
Pham, C.H., and Hancock, G.J. (2009b). "Shear buckling of thin-walled channel sections." Journal
of Constructional Steel Research, 65, 578-85.
Schafer, B.W. (2002). "Local, distortional, and Euler buckling of thin-walled columns." Journal of
Structural Engineering, 128(3), 289-299.
Schafer, B.W., and dny, S. (2006). "Buckling analysis of cold-formed steel members using
CUFSM: conventional and constrained finite strip methods." Eighteenth International
Specialty Conference on Cold-Formed Steel Structures, Orlando, FL.
Southwell, R.V., and Skan, S.W. (1924). "On the stability under shearing forces of a flat elastic
strip." Royal Society -- Proceedings, 105, 582-607.
SSMA. (2001). Product Technical Information, ICBO ER-4943P, Steel Stud Manufacturers
Association, .
Stein, M., and Neff, J. (1947). "Technical Note No. 1222: Buckling stresses of simply supported
rectangular flat plates in shear." NACA, Langley Field, VA.
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Timoshenko, S.P., Gere, James M. (1961). Theory of Elastic Stability, McGraw-Hill, New York,
NY.
Vilnay, O. D. (1990). The behavior of web plate loaded in shear. Thin walled Structures, 10,
161-174
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APPENDIX A
Shear Locking
Shear locking can occur in 2D and 3D solid elements as well as shell
elements. The effect is significant only if there is a certain (in-plane) bending
deformation of the structure. In ideal condition, a plate under pure bending
moment experiences a curved shape as shown in Figure A-1. Under the bending
moment, horizontal dotted lines and edges bend to curves while vertical dotted
lines and edges remain straight. The angle a remains at 90 degrees before and
after bending.
Figure A-1: Shape change of the Block under the Moment in ideal situation
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To correctly model the deformed shape the element should have the ability
to model the curved shape. The edges of some elements are, however, not able to
bend to curves. The linear element will develop a shape as shown in Figure A-2
under a pure bending moment. All dotted lines remain straight. But the angle a
can no longer stay at 90 degrees.
Figure A-2: Shape change of a fully integrated first order element under the Moment
To cause the angle a to change under the pure moment, an incorrect
artificial stress is introduced. This also means that the strain energy of the element
is generating shear deformation instead of bending deformation. The overall
effect is that the linear fully integrated element becomes locked or overly stiff
under the bending moment. Errors in displacement and stresses may be reported
because of the locking.
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Hourglassing
In order to take care of shear locking and to increase computational
efficiency, a reduced integration scheme is used. However the reduced integration
element suffers from its own numerical difficulty called hourglassing (ABAQUS
2008), excessively flexible. Hourglassing results in an element that is excessively
flexible and has to be properly controlled or else the results from this type of
element are often erroneous.
Figure A-3 demonstrates the deformation of such an element under a
bending moment. It can be seen that the vertical and horizontal dotted lines and
the angle a remains unchanged. This implies that the normal stresses and the shear
stresses are zero at the integration point and that there is no strain energy
generated by the deformation. This can admit deformation modes that cause no
straining at the integration points. These zero-energy modes make the element
rank-deficient and cause a phenomenon called hourglassing, where the zero
energy mode starts propagating through the mesh, leading to inaccurate solutions.
This may lead to an error in the results. To prevent these excessive deformations,
an additional artificial stiffness is added to the element. In this so-called hourglass
control procedure, a small artificial stiffness is associated with the zero-energy
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deformation modes. This procedure is used in many of the solid and shell
elements in ABAQUS.
Figure A-3: Shape change of a reduced integrated first order element under the moment M
Membrane Locking
The term membrane locking refers to excessive stiffness in bending. In
this phenomenon the strain fields in the element interact unfavorably, so that
nodal displacements that should be resisted only by bending are resisted by
membrane deformation as well. In other words, as the solid element becomes
thinner, the membrane stiffness starts to dominate, and locking may result.
Common features of all these locking effects is that they lead to parasitic stresses
and thus artificial stiffness in the case of pure bending and that the locking
phenomenon becomes more pronounced as the shell gets thinner. The effects of
membrane locking can be eliminated by the use of reduced stiffness .
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APPENDIX B
The following are the primary MATLAB functions used in the modeling and
calculation of buckling capacity of plates and C-section.
1) Origin: To input data and to call other functions to evaluate the membercapacity.
2) xsect_to_abaqus ( ): Imports model co-ordinates from CUFSM andgenerates co-ordinates to be used in ABAQUS.
3) nodeset ( ): Creates a nodeset based on user defined conditions.4) specgeom ( ): converts the out-to-out dimensions into xy-co-ordinates to
be used in CUFSM.
5) node_element_matrices ( ): Creates the node and element matrix to beused in ABAQUS.
6) loadgeneration ( ): Creates a load matrix.7) inputgeneration ( ): Generates an ABAQUS input file (*.inp).8) grabber ( ): Searches an ABAQUS .dat file for eigen values and places
them in a matrix.
9) infernoscript ( ): Creates a PBS queing script file to submit anABAQUS.inp file to inferno2 (Virginia Tech) for processing.
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10)submitscript ( ): Creates a Linux script file which can be used to submitmultiple script file to Inferno2 (Virginia Tech).
The code for each of the functions is provided below:
1) Origin:clear allclose allglobal Ey;Ex=29500;Ey=29500;vx=0.3;vy=0.3;G=11000;L=40;
filename='ex01';load(filename);
W=(node(15,3))W=abs(W);nele=80;
savename='plateelementex01';
elemtype='S9R5';analysis=1;mat=100;matprops=[mat Ex Ey vx vy G];
[FEnode,FEelem,t,matnum,nnodes,nL,FEsection_increment,elemgroups,node]=xsect_to_abaqus(filename,L,nele,analysis);
noderange=[0 L 0 0 -W -W];[nodeloc1]=nodeset (FEnode, noderange);
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nodenum1=FEnode(nodeloc1,1);
noderange=[0 L 0 0 W W];[nodeloc2]=nodeset (FEnode, noderange);nodenum2=FEnode(nodeloc2,1);
noderange=[0 0 0 0 -W W];[nodeloc3]=nodeset (FEnode, noderange);nodenum3=FEnode(nodeloc3,1);
noderange=[L L 0 0 -W W];[nodeloc4]=nodeset (FEnode, noderange);nodenum4=FEnode(nodeloc4,1);
noderange=[0 0 0 0 -W -W];nodeloc5=nodeset (FEnode, noderange);nodenum5=FEnode(nodeloc5,1);
%new addednoderange=[L L 0 0 W W];nodeloc6=nodeset (FEnode, noderange);nodenum6=FEnode(nodeloc6,1);
nodelocA=[nodenum1 nodenum2 ];nodelocB=[nodenum3 nodenum4 ];nodelocperimA=unique(nodelocA);
nodelocperimB=unique(nodelocB);
[cloadele1, cloadele2, cloadele3,cloadele4]=loadgeneration(L,W,nodenum1,nodenum2,nodenum3,nodenum4,t(1));
inputgeneration(FEnode,FEelem,savename,matprops,elemtype,matnum,mat,cloadele1,cloadele2,cloadele3,cloadele4,t,nodelocA,nodelocB,nodenum5,nodenum6,analysis);
walltime=[0 1];cpus=[1];
scriptname='script';
infernoscript(jobname,walltime,cpus);
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submitscript(scriptname, jobname);
end
2) xsect_to_abaqus ( ) :function[FEnode,FEelem,t,matnum,nnodes,nL,FEsection_increment,elemgroups,node]=xsect_to_abaqus(filename,L,nele,analysis)%
%cufsm_to_abaqus.m%Ben Schafer%December 2005%***********%Modified by Cris Moen%November 2006%Notes: modified cufsm_to_abaqus for use as bare bones S9R5 nodeand element generator
%Round nodal coordinates to elimate accuracy noise%node(:,2:3)=round(node(:,2:3)*1000)/1000;
%WARNINGS%Has to be an even number of FSM elements for this to work% if rem(length(elem(:,1)),2)>0% ['Warning! Your CUFSM model has an odd number of elementsthis will not convert to ABAQUS S9R5 elements. Please modify yourmodel so that the number of elements is an even number']% end%load(filename);%PRELIMINARIES%Count FSM nodes and modesnnodes=length(node(:,1));%Number of FSM cross-section nodes%nmodes=length(curve(:,1)); %Number of FSM mode shapes for firstmode, same as number of lengths%Determine FE number of nodes and increment
nL=2*nele+1; %Number of FE nodes along the length%Determine the node numbering increment along the lengthif nnodes
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FEsection_increment=100; %so along the length the numberinggoes up by 100'selse
FEsection_increment=nnodes+1;end
%NODAL COORDINATES IN FE FORM
undefx=zeros(nnodes,nL);undefz=zeros(nnodes,nL);for i=1:nL
undefx(:,i)=node(:,2);undefz(:,i)=node(:,3);
undefy(:,i)=ones(nnodes,1)*(i-1)/(nL-1)*L;end%Define variable for the deformed/