EnergyBased Approach for Buckling Problems in Steel Structures

1
Energy-Based Approach for Buckling Problems in
Steel Structures
NATIONAL RESEARCH LABORATORY
October 12, 2000
Jaehong Lee Dept. of Architectural
Engineering Sejong University
2
OBJECTIVES
To Present Energy Method in Buckling Analysis
Structural Behavior of Cold-formed Channel
Section Beams
State-of-the Art Review of the Analysis of
Thin-walled Structures
3
CONTENTS

• Introduction
• Impact of cold-formed steel
• Structural Consideration of Channel section
• Lateral Buckling
• Flexural-Torsional Buckling
• Stress Analysis
• Local Buckling Effective Width
• Analysis Design of Cold-formed Channel
• Next Steps

4
COLD-FORMED STEEL OFFERS VERSATILITY IN BUILDINGS
Cold-formed steel represents over 45 percent of
the steel construction market in
U.S. Sophisticated structures such as schools,
churches and complex manufacturing facilities.

• Ease of Prefabrication and Mass Production

Light Weight
Uniform Quality
Economy in Transformation and Handling
Quick and simple erection
5
ANALYSIS DESIGN OF COLD-FORMED CHANNEL-SECTION
BEAMS ARE NOT EASY
How to Take Care of These Complecated
Behavior Finite Element Analysis AISI Aisi
Code Effective Width Linear Method Iterative
Method Bending Torsion
Things to Consider in Analysis and Design of
Beams Elastic Lateral Buckling Inelastic Lateral
Buckling Local Buckling Sectional
Properties Center of Gravity ? Shear Center
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CONTENTS

• Introduction
• Impact of cold-formed steel
• Structural Consideration of Channel section
• Lateral Buckling
• Flexural-Torsional Buckling
• Stress Analysis
• Local Buckling Effective Width
• Analysis Design of Cold-formed Channel
• Next Steps

7
LATERAL BUCKLING MAY OCCUR WELL BELOW THE YIELD
STRENGTH LEVEL
Original position
u
v
Fy
Elastic Lateral Buckling Strength
Final position for inplane bending
?
?
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FINITE ELEMENT MODEL IS THE BEST
Kinematics
Variational Formulation
Constitutive Relations
Lateral Buckling Equations
Finite Element Model

• Build the appropriate displacement fields
• Derive the strain tensor

• Kinematics

Variational Formulation

• Strain energy
• Potential of transverse load at shear center

Constitutive Relations

• Stress resultants vs. strains

Lateral Buckling Equations

• Can be derived by integrating by parts
• Coupled differential equations

Finite Element Model

• Setup the eigenvalue problem
• Buckling loads and mode shapes

9
KINEMATICS OF THIN-WALLED SECTION
Basic Assumptions
Contour Coordinate

• Kirchhoff-Love assumption
• Shear strain at midsurface is zero.

Displacement Field
Plate Action
Beam Action
10
VARIATIONAL FORMULATION IS USED TO FORMULATE THE
GOVERNING EQUATIONS

• WEAK FORM

a
CONSTITUTIVE MODEL
s.c
Load Type
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GOVERNING LATERAL BUCKLING EQUATIONS CAN BE
DERIVED BY INTEGRATION BY PARTS THE VARIED
QUATITIES
Lateral Buckling Equations
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FINITE ELEMENT MODEL IS DERIVED FROM THE WEAK FORM
Finite Element Model (Standard Eigenvalue Problem)
? eigenvalue (buckling parameter) ?
eigenfunction (buckling mode shape)
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CLOSED-FORM SOLUTION FOR ELASTIC LATERAL BUCKLING
IS LIMITED
Simply-supported Beam Under Pure Bending
M
M
Buckling stress
Buckling moment
Buckling mode shape
(For H-section)
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UNEQUAL END MOMENTS AND VARIOUS BOUNDARY
CONDITIONS SHOULD BE CONSIDERED
Bending coefficient (moment gradient factor) Cb
M1
(?)
M2
M2 gtM1 M gtM2 Cb1
M1
()
M2
AISI Specification
1968-1980 edition St. Venant torsion neglected
1989 edition Pekoz Winter For singly-symmetric
section torsional-flexural buckling considered
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BEAM UNDER UNEQUAL END MOMENTS
M
bM
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BUCKLING MODES OF A BEAM UNDER UNEQUAL END MOMENTS
M
bM
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EFFECT OF LOADING POINT ON A CANTILEVER BEAM
UNDER POINT LOAD AT FREE END
P
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EFFECT OF LOADING POINT ON A SIMPLY-SUPPORTED
BEAM UNDER UNIFORMLY-DISTRIBUTED LOAD
w
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LATERALLY-SUPPORTED BEAM UNDER UNEQUAL END MOMENTS
M
bM
x
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LATERALLY-SUPPORTED BEAM UNDER UNEQUAL END MOMENTS
M
bM
x
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INELASTIC LATERAL BUCKLING SHOULD BE CONSIDERED
FOR REAL PROBLEMS
When buckling stress exceeds the proportional
limit
Inelastic Lateral Buckling
Elastic Lateral Buckling
Fy
The beam behavior is governed by inelastic
buckling
spr
For accurate solution, rigorous iterative method
is required
?
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AISI CODE PROVIDES CONSERVATIVE INELASTIC
BUCKLING MOMENT
Mcr/My
1.0
0.5
My/Me
1
3
2
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CONTENTS

• Introduction
• Impact of cold-formed steel
• Structural Consideration of Channel section
• Lateral Buckling
• Flexural-Torsional Buckling
• Stress Analysis
• Local Buckling Effective Width
• Analysis Design of Cold-formed Channel
• Next Steps

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SAME PROCEDURE EXCEPT THE WORK DONE BY FORCES
Kinematics
Variational Formulation
Constitutive Relations
Lateral Buckling Equations
Finite Element Model

• Build the appropriate displacement fields
• Derive the strain tensor

• Kinematics

Variational Formulation

• Strain energy
• Potential of external forces

Constitutive Relations

• Stress resultants vs. strains

Lateral Buckling Equations

• Can be derived by integrating by parts
• Coupled differential equations

Finite Element Model

• Setup the eigenvalue problem
• Buckling loads and mode shapes

25
VARIATIONAL FORMULATION IS USED TO FORMULATE THE
GOVERNING EQUATIONS

• WEAK FORM

CONSTITUTIVE MODEL
s.c
c.g
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GOVERNING FLEXURAL-TORSIONAL BUCKLING EQUATIONS
CAN BE DERIVED BY INTEGRATION BY PARTS THE VARIED
QUATITIES
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FINITE ELEMENT MODEL IS DERIVED FROM THE WEAK FORM
Finite Element Model (Standard Eigenvalue Problem)
? eigenvalue (buckling parameter) ?
eigenfunction (buckling mode shape)
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CONTENTS

• Introduction
• Impact of cold-formed steel
• Structural Consideration of Channel section
• Lateral Buckling
• Flexural-Torsional Buckling
• Stress Analysis
• Local Buckling Effective Width
• Analysis Design of Cold-formed Channel
• Next Steps

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WHEN THE TRANSVERSE LOADS DO NOT PASS THROUGH THE
SHEAR CENTER, THE MEMBER WILL BE SUBJECTED TO
BOTH BENDING AND TORSION
Bending
Lateral Buckling

• Loads applied at shear center

s.c
v
Bending Torsion
c.g
Loads applied at center of gravity
f
v
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VARIOUS NORMAL AND SHEAR STRESSES CAN BE
GENERATED

• BENDING

• Longitudinal bending stress
• Shear stress

TORSION

• Warping longitudinal stress
• Pure torsional shear stress
• Warping shear stress

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WARPING CHARACTERISTICS OF CHANNEL SECTION
y
Shear Center Location
t
Definition
x
c.g
s.c
d
xp
Channel Section
b
?n3
Normalized Unit Warping
?n4
Definition
Channel Section
?n2
?n1
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WARPING CHARACTERISTICS OF CHANNEL SECTION
Warping Moment of Inertia
Definition
Channel Section
b3
Warping Static Moment
S?3
S?4
Definition
Channel Section
S?5
S?1
S?2
S?6
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STRESS ANALYSIS OF CHANNEL SECTION BEAM
EXAMPLE PROBLEM
0.3k/ft
1.5
7
10
0.135
Load applied at shear center
Load applied at center of gravity
-
-
Normal stress



?b
?w
?b
Shear stress


?v
?w
?v
?t
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RESULTS OF STRESS ANALYSIS EXAMPLE PROBLEM
A member exhibiting bending-torsion coupling
shows significantly different stress distribution
3
4
5
1
2
6
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CONTENTS

• Introduction
• Impact of cold-formed steel
• Structural Consideration of Channel section
• Lateral Buckling
• Stress Analysis
• Local Buckling Effective Width
• Analysis Design of Cold-formed Channel
• Next Steps

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LOCAL BUCKLING CAN OCCUR BEFORE GLOBAL BUCKLING
Reduce the ultimate load-carrying capacity
significantly
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BEHAVIOR OF STIFFENED AND UNSTIFFENED
COMPRESSION ELEMENTS ARE NOT IDENTICAL
Stiffened compression elements (s.c.e)
A flat compression elements stiffened by other
components (web, flange, lip, stiffener) along
both longitudinal edges
Unstiffened compression elements (u.c.e)
A flat compression element stiffened only along
one of the two longitudinal edges
u.c.e
u.c.e
s.c.e
s.c.e
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PLATES DO NOT COLLAPSE WHEN BUCKLING OCCURS, BUT
CAN STILL CARRY LOAD AFTER BUCKLING -
POSTBUCKLING STRENGTH
p
pcr
d
Plate Buckling Equation
Plate Buckling Stress

• Rigorous solution of postbuckling is difficult
  (Nonlinear numerical Analysis needed)
• Can define EFFECTIVE width

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EFFECTIVE DESIGN WIDTH b CONCEPT IS WIDELY USED
IN DESIGN PROCEDURE DUE TO THEIR SIMPLICITY
First introduced by von Karman (1932)
effective width, b, represents a width of the
plate which just buckles when ? ?y
The initially uniform compressive stresses become
redistributed
?max
Relation of b and w
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AISI SPECIFICATION FOR EFFECTIVE WIDTH HAS BEEN
DEVELOPED
AISI design provision (1946-1968)
Winter (1946) presented the formula for effective
width
AISI design provision (1970- )
Winter (1970) presented more realistic equation
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AISI DESIGN PROVISION FOR EFFECTIVE WIDTH
Effective Design Width Equation
Individual plates subjected to different boundary
conditions
Need to calculate k
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BUCKLING STRESSES CAN BE DETERMINED VIA
COEFFICIENT K
s.s.
s.s.
s.s.
s.s.
s.s.
s.s.
s.s.
s.s.
fixed
fixed
s.s.
s.s.
fixed
fixed
fixed
fixed
s.s.
s.s.
s.s.
s.s.
s.s.
s.s.
free
s.s.
fixed
fixed
s.s.
s.s.
fixed
fixed
free
fixed
fixed
s.s.
s.s.
s.s.
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CALCULATION OF EFFECTIVE WIDTH OF COMPRESSION
FLANGE FOR CHANNEL IS STRAIGHTFORWARD
Check if
Check the width-to-thickness ratio
k0.425
Buckling coefficient for ss-ss-ss-free
Calculation of slenderness ratio
Calculation of efffective width parameter
Determine the effective width
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EFFECTIVE WIDTH OF WEB SHOULD BE CALCULATED BY
ITERATION PROCESS (NOT SIMPLE)
w
Assume fully effective
b
f1
b1
Check if
hc
b2
Recalculate the neutral axis
f2
no
n1
ngt1
no
Check if
yes
yes
Web is fully effective!
b1 b2 calculated
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CALCULATION OF EFFECTIVE WIDTH OF COMPRESSION
FLANGE FOR LIPPED CHANNEL IS DEPENDENT TO THE
RIGIDITY OF THE LIP
w
and
No edge stiffnener needed
Check if
for
for
for lip stiffener
ds
D
ds
d
for
for
for lip stiffener
For edge stiffener k0.425
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ANALYSIS AND DESIGN OF COLD-FORMED STEELS ARE
INTEGRATED PROCEDURE
Ideas for local buckling
Ideas for lateral buckling
Ideas for Inelastic buckling
Ideas for stress analysis
Design
Analysis
AISI code
FEM
Stresses
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CONTENTS

• Introduction
• Impact of cold-formed steel
• Structural Consideration of Channel section
• Lateral Buckling
• Stress Analysis
• Local Buckling Effective Width
• Analysis Design of Cold-formed Channel
• Next Steps

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DESIGN STRENGTH CAN BE CALCULATED VIA COMPLICATED
PROCEDURE
Sectional Properties
Calculate the sectional properties (A, x, y, S,
J, Ix ,Iy ,Iw) of full section by linear method
Elastic Lateral Buckling Moment
Determine buckling moment and mode using accurate
finite element analysis or AISI code
Inelastic Lateral Buckling Moment
Determine inelastic buckling moment using AISI
code
Effective Width of Flange and Lip
Determine the effective width of compression
flange and edge stiffener
Effective Width of Web
Assume fully effective web and check the
effectiveness by iteration
Effective Sectional Modulus
Recalculate the neutral axis until the effective
web width is determined
Nominal and Design Strength
The interaction of the local and overall lateral
buckling results in a reduction of the lateral
strength
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DESIGN STRENGTH OF CHANNEL BEAM - EXAMPLE PROBLEM
Sectional Properties
P
By linear method
Elastic Critical Moment is calculated from AISI
code or FEM
Inelastic Critical Moment is calculated from AISI
code
(4 reduction)
Nominal Moment is based on the effective
sectional modulus
(31 reduction)
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RIGOROUSLY ANALYSE THE TECHNOLOGY TREE OF
COLD-FORMED STEEL MEMBER
Flexural Members
Bending
Bending Strength
Effective Sectional Prop.
Stress Analysis
Local Buckling
Pure Torsion
Lateral Buckling
Deflection
Warping
Beam Webs
Bending Torsion
Purlins
Distortional Buckling
COLD-FORMED STEEL MEMBER
Compressive Members
Compressive Strength
Flexural Buckling
Cold Work
Cylindrical tubular members
Torsional Buckling
Effective Length
Local Buckling
Wall Studs
Shear Diaphragms
other cross-sections
Corrugated Sheets
Composite Design
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CONTENTS

• Introduction
• Impact of cold-formed steel
• Structural Consideration of Channel section
• Lateral Buckling
• Stress Analysis
• Local Buckling Effective Width
• Analysis Design of Cold-formed Channel
• How do we take care of the combined effects?

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VARIOUS TYPES OF BUCKLING CAN OCCUR
Local Buckling Each plate element can buckle
Distortional Buckling Lateral deflection of the
unsupported flange

• Global Buckling
• profile of cross section does not change

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DIFFERENT AVAILABLE NUMERICAL METHODS

• Plate Finite Elements
• Finite Strip Method
• Beam Models
• Effective Width Concept
• Special Constitutive Law
• Enriched displacement field
• Plate FE with static condensation of d.o.f.s

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PLATE FINITE ELEMENTS

• Can model local effects
• Requires a fine mesh
• Practical Difficulties

55
FINITE STRIP METHOD

• D.O.F.s can be reduced
• Limited to prismatic simply-supported members
  with constant forces

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BEAM MODELS

• Good for overall stability
• Nondeformability of the profile cross section
• Cannot account for local effects

• Effective Width Concept
• limited to local buckling
• Try to represent the effect rather than the
  phenomenon itself
• Enriched displacement field
• Local deformation of the cross section is
  superimposed in a displacement field
• Assumed that the shape of the local field is
  unchanged during the process
• Plate FE with static condensation of d.o.f.s
• Modeled as plate finite elements with restrained
  d.o.f.
• Classical beam d.o.f. magnitude of the local
  deformation
• Timoshenko beam model
• Transverse shear deformation

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CONCLUDING REMARKS

• The geometric coupling depends on the shape of
  the cross section.
• Needs fully geometrically nonlinear model to
  predict the structural behavior accurately.
  (tremendous efforts)
• Beam model seems reasonable, but can be improved
  by considering local effects or shear
  deformation.
• Consideration of material nonlinearity including
  inelastic buckling can be achieved by stress
  analysis or global assumption of plastic process.
• Structural members with anisotropic materials
  (pultruded composites) awaits future attention.

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Thank you