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