Michael H. Swanger Georgia Tech CASE Center June, 2011
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Transcript of Michael H. Swanger Georgia Tech CASE Center June, 2011
GTStrudl Training…
Nonlinear Geometric Analysis of
Structures…
Some Practical Fundamentals and Insights
Michael H. Swanger
Georgia Tech CASE CenterJune, 2011
• Lite Overview of Basic Concepts- Equilibrium Formulation- Element Nodal Forces- Element Implementation Behavior Assumptions- Tangent Stiffness
• Simple Basic behavior Examples- Simply-supported beam under axial load, imperfect
geometry- Shallow truss arch: snap-through behavior- Shallow arch toggle: SBHQ6 model, snap-through
behavior- Slender cantilever shear wall under axial load -- in-
plane SBHQ plate behavior- The P-δ Question!
• Additional Examples
Topics
2GTSUG, 2011, Delray Beach,FLJune 22-25, 2011
Overview of Basic Concepts
The Principle of Virtual Work :
( ) ( ) 0
( ) ( ) 0
( ) ( ) 0
T T
T
u u dV P u
u B u u dV P u
B u u dV P
Equilibrium Formulation
3GTSUG, 2011, Delray Beach,FLJune 22-25, 2011
Overview of Basic Concepts
3 31 1 2 21
2
The Element Equation of Equilibrium :
( ) ( ) 0
( ) ( )
( ) { }
{ ( )} { } 0
ij
T
T T TL NL
jiL NL
j i j j j j j j
T TL NL L NL
B u u dV P
B u B B u
uu u uu u u uu D
x x x x x x x x
B B u D dV P
Equilibrium Formulation
4GTSUG, 2011, Delray Beach,FLJune 22-25, 2011
The Equation of Element Equilibrium -- Element Nodal Forces :
{ ( ) ( ) }
[ ]{ }, [ ( )]{ }
{[ ]{ }
[ ( )]{ } [ ( ) ]{ }
[ ( ) ( )]{ }}
T T T TL L L NL NL L NL NL
L L NL NL
TL L
T TL NL NL L
TNL NL
B D B D B u D B u D dV P
B u G u u
B DB u
B DG u u B u DB u
B u DG u u dV P
Overview of Basic ConceptsElement Nodal Forces
5GTSUG, 2011, Delray Beach,FLJune 22-25, 2011
Overview of Basic ConceptsElement Implementation Behavior Assumptions
Assumptions related to the scope of nonlinear geometricbehavior are introduced into the definition of strain and the equilibrium equation:
2 2
2 2
22 2
2 2
2 2
[ ( )]{ } [ ( ) ]
1
2
{[ ]{ }
{
[ ( )
}
]{( ) }}
yx zx
TL L
yx z
TNL N
y z
T TL NL
L
NL L
uu uy z
x x x
B DB u
u
uu uy z
x x x
B u
u u
x x
B DG u u B u D
u dVDG u P
B
Example: Frame Member Strain and Equilibrium
6GTSUG, 2011, Delray Beach,FLJune 22-25, 2011
0
Axial Transverse
Torsion Transverse
P and M are coupled
Modified U and U are uncoupled
θ and U are uncoupled
0
Overview of Basic ConceptsElement Implementation Behavior Assumptions
Summary of GTSTRUDL NLG Behavior Assumptions
1. Plane and Space Frame
− Small strains; σ = Eε remains valid− Internal rotations and curvatures are small; θ ≈ sinθ− Member chord rotations are small− P and M are coupled− Uaxial and UTransverse are uncoupled− θTorsion and UTransverse are uncoupled− Other member effects are not affected by member displacement− Member loads are not affected by member displacement
2. Plane and Space Truss
− Small strains; σ = Eε remains valid− No assumptions limiting magnitude of displacements
7GTSUG, 2011, Delray Beach,FLJune 22-25, 2011
Overview of Basic ConceptsElement Implementation Behavior Assumptions
Summary of GTSTRUDL NLG Behavior Assumptions
3. SBHQ and SBHT Plate Elements
− Small strains; σ = Dε remains valid− BPH + PSH + 2nd order membrane effects
Internal rotations and curvatures are smallUin-plane and UTransverse are coupled in 2nd order membrane effectsBPH and 2nd order membrane effects are uncoupled
− Element loads are not affected by element displacements
4. The IPCABLE Element
− Small strains; σ = Eε remains valid− No assumptions limiting magnitude of displacements− Regarding NLG, 2-node version and the truss are the same
8GTSUG, 2011, Delray Beach,FLJune 22-25, 2011
Overview of Basic Concepts
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
Incremental Equation of Element Equilibrium:
0
,
;
TL NL L NL
TL NL L NL
T TL NL L NL L L
T
NL L N
u
B dV P
d B dV u P where du
dB dV B d dV u
u P
P
K K u P K
The Tangent Stiffness Matrix
9GTSUG, 2011, Delray Beach,FLJune 22-25, 2011
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 10
Overview of Basic ConceptsThe Tangent Stiffness Matrix
u
P
Pi
Pi+1
ui ui+1
a
1
b
2
u1 u2
u1=ui+u1
u2=u1+u2
KT = [Kσ + Ku] TB σdV
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 11
• Simply-supported beam under axial load, imperfect geometry
• Shallow truss arch: snap-through behavior
• Shallow arch toggle: SBHQ6 model, snap-through behavior
• Slender cantilever shear wall under axial load -- in-plane SBHQ plate behavior
• The P-δ Question!
Simple Basic behavior Examples
12GTSUG, 2011, Delray Beach,FLJune 22-25, 2011
Simple Basic Behavior ExamplesSimply-supported beam under axial load, imperfect geometry
20 @ 1 ft
Imperfection: Yimp = -0.01sin(πx/L) ft
P
E = 10,000 ksiPlane Frame: Ax = 55.68 in2, Iz = 100.00 in4
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 13
Simple Basic Behavior ExamplesSimply-supported beam under axial load, imperfect geometry
Pe = 171.2 kips
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 14
Simple Basic Behavior ExamplesSimply-supported beam under axial load, imperfect geometry
Push-over Analysis Procedure
UNITS KIPSLOAD 1JOINT LOADS 21 FORCE X -1000.0 $ Load P
NONLINEAR EFFECTS GEOMETRY MEMBERS EXISTING
PUSHOVER ANALYSIS DATA INCREMENTAL LOAD 1 MAX NUMBER OF LOAD INCR 200 MAX NUMBER OF TRIALS 20 MAX NUMBER OF CYCLES 100 LOADING RATE 0.005 $ f1
CONVERGENCE RATE 0.8 CONVERGENCE TOLERANCE COLLAPSE 0.0001 CONVERGENCE TOLERANCE DISPLACEMENT 0.001ENDPERFORM PUSHOVER ANALYSIS
f1P
Displacement
Load P
1
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 15
Simple Basic Behavior ExamplesSimply-supported beam under axial load, imperfect geometry
Push-over Analysis Procedure
UNITS KIPSLOAD 1JOINT LOADS 21 FORCE X -1000.0
NONLINEAR EFFECTS GEOMETRY MEMBERS EXISTING
PUSHOVER ANALYSIS DATA INCREMENTAL LOAD 1 MAX NUMBER OF LOAD INCR 200 MAX NUMBER OF TRIALS 20 MAX NUMBER OF CYCLES 100 LOADING RATE 0.005 $ f1
CONVERGENCE RATE 0.8 CONVERGENCE TOLERANCE COLLAPSE 0.0001 CONVERGENCE TOLERANCE DISPLACEMENT 0.001ENDPERFORM PUSHOVER ANALYSIS
f1P
(2f1)P
Displacement
Load P
1
2
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 16
Simple Basic Behavior Examples
Push-over Analysis Procedure
UNITS KIPSLOAD 1JOINT LOADS 21 FORCE X -1000.0
NONLINEAR EFFECTS GEOMETRY MEMBERS EXISTING
PUSHOVER ANALYSIS DATA INCREMENTAL LOAD 1 MAX NUMBER OF LOAD INCR 200 MAX NUMBER OF TRIALS 20 MAX NUMBER OF CYCLES 100 LOADING RATE 0.005 $ f1
CONVERGENCE RATE 0.8 CONVERGENCE TOLERANCE COLLAPSE 0.0001 CONVERGENCE TOLERANCE DISPLACEMENT 0.001ENDPERFORM PUSHOVER ANALYSIS
Simply-supported beam under axial load, imperfect geometry
f1P
(2f1)P
(3f1)P
Displacement
Load P
1
3
2
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 17
Simple Basic Behavior Examples
Push-over Analysis Procedure
UNITS KIPSLOAD 1JOINT LOADS 21 FORCE X -1000.0
NONLINEAR EFFECTS GEOMETRY MEMBERS EXISTING
PUSHOVER ANALYSIS DATA INCREMENTAL LOAD 1 MAX NUMBER OF LOAD INCR 200 MAX NUMBER OF TRIALS 20 MAX NUMBER OF CYCLES 100 LOADING RATE 0.005 $ f1
CONVERGENCE RATE 0.8 $ r CONVERGENCE TOLERANCE COLLAPSE 0.0001 CONVERGENCE TOLERANCE DISPLACEMENT 0.001ENDPERFORM PUSHOVER ANALYSIS
Simply-supported beam under axial load, imperfect geometry
f1P
(2f1)P
(3f1)P
Displacement
Load P
(2f1 + rf1)P
1
3
4
2
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 18
Simple Basic Behavior ExamplesShallow truss arch: snap-through behavior
3 in
u
3 - u
2 @ 100 in
L
L’
θ
P
E = 29,000 ksiPlane Truss: Ax = 1.0 in2
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 19
Simple Basic Behavior ExamplesShallow truss arch: snap-through behavior
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 20
Simple Basic Behavior ExamplesShallow arch toggle: SBHQ6 model, snap-through behavior
2 @ 12.943 in
0.3667 in
X
Y E = 1.0300000E+07 lbs/in2
ν = 0.0
Fixed (typ)
P
A
A
0.753 in0.243 in
Section A-A
SBHQ6 Arch Leg, 20 x 4
Θz = 0
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 21
Simple Basic Behavior ExamplesShallow arch toggle: SBHQ6 model, snap-through behavior
Note: Pbuck = 152.4 lbs (linear buckling load)
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 22
Slender cantilever shear wall under axial load -- in-plane SBHQ plate behavior
Simple Basic Behavior Examples
0.01 kips
P
Mesh = 2X50Material = concrete
POISSON = 0.0Thickness = 4 in
100 ft
2 ft
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 23
Slender cantilever shear wall under axial load -- in-plane SBH plate behavior
Simple Basic Behavior Examples
Pbuck (FE) = 41.95 kips
(Pe (SF) = 28.42 kips)
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 24
The P-δ Question
Does GTSTRUDL Include P-δ?
E = 10,000 ksi, Plane Frame: Ax = 55.68 in2, Iz = 100.0 in4
No Mid Span Nodes
1 Mid Span Node
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 25
The P-δ Question
June 22-25, 2011 GTSUG, 2011, Delray Beach,FL 26
The P-δ Question
Mtot = M0 + Pδmid