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



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


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


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


0

Axial Transverse

Torsion Transverse

P and M are coupled

Modified U and U are uncoupled

θ and U are uncoupled

0


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



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



( ) ( )

( ) ( )

( ) ( ) ( ) ( )

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


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


• 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


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



Pe = 171.2 kips



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




UNITS KIPSLOAD 1JOINT LOADS 21 FORCE X -1000.0




f1P

(2f1)P

Displacement

Load P

1

2


Simple Basic Behavior Examples






Simply-supported beam under axial load, imperfect geometry

f1P

(2f1)P

(3f1)P

Displacement

Load P

1

3

2







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


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


Simple Basic Behavior ExamplesShallow truss arch: snap-through behavior


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


Simple Basic Behavior ExamplesShallow arch toggle: SBHQ6 model, snap-through behavior

Note: Pbuck = 152.4 lbs (linear buckling load)


Slender cantilever shear wall under axial load -- in-plane SBHQ plate behavior


0.01 kips

P

Mesh = 2X50Material = concrete

POISSON = 0.0Thickness = 4 in

100 ft

2 ft


Slender cantilever shear wall under axial load -- in-plane SBH plate behavior


Pbuck (FE) = 41.95 kips

(Pe (SF) = 28.42 kips)


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


The P-δ Question


The P-δ Question

Mtot = M0 + Pδmid

michael h. swanger georgia tech case center june, 2011

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