naik shearbuckling

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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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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_bin


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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.
http://en.wikipedia.org/wiki/ANSIhttp://en.wikipedia.org/wiki/ANSI


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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/Providence


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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/

naik shearbuckling

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