# GTStrudl Modeling and Analysis Using 2-D and 3-D Finite Elements by qingyunliuliu

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```									            GTStrudl Training
…
Nonlinear Geometric Analysis
of
Structures
…
Some Practical Fundamentals and Insights

Michael H. Swanger

Georgia Tech CASE Center
June, 2011
Topics
•     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!

June 22-25, 2011          GTSUG, 2011, Delray Beach,FL         2
Overview of Basic Concepts
Equilibrium Formulation

The Principle of Virtual Work :

  (u )  (u ) dV  P u  0

 u T   BT (u ) (u ) dV  P   u  0
                        

 BT (u ) (u ) dV  P  0

June 22-25, 2011             GTSUG, 2011, Delray Beach,FL     3
Overview of Basic Concepts
Equilibrium Formulation

The Element Equation of Equilibrium :

 BT (u ) (u ) dV  P  0

BT (u )  BL  BNL (u )
T    T

       1  ui u j  u1 u1 u2 u2 u3 u3   
 (u )  D{ L   NL }            ij                                      
       2  x j        x x x x x x   
xi  j j                   
                               j   j   j   j  

 {BL  BNL (u )}D{ L   NL }dV  P  0
T    T

June 22-25, 2011             GTSUG, 2011, Delray Beach,FL                  4
Overview of Basic Concepts
Element Nodal Forces

The Equation of Element Equilibrium -- Element Nodal Forces :

 {BL D L  BL D NL  BNL (u ) D L  BNL (u ) D NL } dV  P
T         T          T               T

 L  [ BL ]{u},       NL  [GNL (u )]{u}

 {[BL DBL ]{u} 
T

[ BL DGNL (u )]{u}  [ BNL (u ) DBL ]{u} 
T                    T

[ BNL (u ) DGNL (u )]{u}} dV  P
T

June 22-25, 2011               GTSUG, 2011, Delray Beach,FL                    5
Overview of Basic Concepts
Element Implementation Behavior Assumptions

Assumptions related to the scope of nonlinear geometric
behavior are introduced into the definition of strain and
the equilibrium equation:

Example: Frame Member Strain and Equilibrium

u    u     u
2
1  u  u
22
u
2    u y   uz 2 
2       2

 x  x  y 2y  z 2z   x  y 2y  z 2z   
 x                          
x    x     x   2      x     x   x   x  
                                     
0                 P and M are coupled                   
                                      
Modified UAxial and U Transverse are uncoupled 
 {[BL DBL ]{u} 
T                                                     θ                                     
 Torsionand U Transverse are uncoupled

[ BL DGNL (u )]{u}  [ BNL (u ) DBL ]{u} 
T                    T
0

[ BNL (u ) DGNL (u )]{u}} dV  P
T

June 22-25, 2011                     GTSUG, 2011, Delray Beach,FL                                         6
Overview of Basic Concepts
Element 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
June 22-25, 2011                GTSUG, 2011, Delray Beach,FL             7
Overview of Basic Concepts
Element 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 small
Uin-plane and UTransverse are coupled in 2nd order membrane effects
BPH 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

June 22-25, 2011                GTSUG, 2011, Delray Beach,FL                8
Overview of Basic Concepts
The Tangent Stiffness Matrix

Incremental Equation of Element Equilibrium:

 B(TL  NL ) ( L  NL ) dV  P  0

d    B T
( L  NL )          
 ( L  NL ) dV u  P, where d 

u

 dB(TL  NL ) ( L  NL ) dV  B(TL  NL ) d ( L  NL ) dV  u  P
                                                          

 K  Ku  u  P;
                                     KT u  P
June 22-25, 2011                  GTSUG, 2011, Delray Beach,FL                     9
Overview of Basic Concepts
The Tangent Stiffness Matrix

P

KT = [Kσ + Ku]
 B σdV
T

a          b
Pi+1
2
1
u2=u1+u2
Pi
u1         u2

ui    u1=ui+u1                       ui+1       u
June 22-25, 2011                GTSUG, 2011, Delray Beach,FL                  10
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!

June 22-25, 2011           GTSUG, 2011, Delray Beach,FL             11
Simple Basic Behavior Examples
Simply-supported beam under axial load, imperfect geometry

P

20 @ 1 ft

Imperfection: Yimp = -0.01sin(πx/L) ft

E = 10,000 ksi
Plane Frame:   Ax = 55.68 in2,  Iz = 100.00 in4

June 22-25, 2011                    GTSUG, 2011, Delray Beach,FL                      12
Simple Basic Behavior Examples
Simply-supported beam under axial load, imperfect geometry

Pe = 171.2 kips

June 22-25, 2011        GTSUG, 2011, Delray Beach,FL             13
Simple Basic Behavior Examples
Simply-supported beam under axial load, imperfect geometry

Push-over Analysis Procedure

UNITS KIPS
21 FORCE X -1000.0     \$ Load P

GEOMETRY MEMBERS EXISTING

PUSHOVER ANALYSIS DATA
MAX NUMBER OF LOAD INCR 200
MAX NUMBER OF TRIALS 20
MAX NUMBER OF CYCLES 100
CONVERGENCE RATE     0.8
CONVERGENCE TOLERANCE COLLAPSE 0.0001
CONVERGENCE TOLERANCE DISPLACEMENT 0.001
END
1
PERFORM PUSHOVER ANALYSIS
f1P

Displacement

June 22-25, 2011                   GTSUG, 2011, Delray Beach,FL             14
Simple Basic Behavior Examples
Simply-supported beam under axial load, imperfect geometry

Push-over Analysis Procedure

UNITS KIPS
21 FORCE X -1000.0
NONLINEAR EFFECTS
GEOMETRY MEMBERS EXISTING

PUSHOVER ANALYSIS DATA
MAX NUMBER OF LOAD INCR 200
MAX NUMBER OF TRIALS 20
MAX NUMBER OF CYCLES 100
CONVERGENCE RATE     0.8
CONVERGENCE TOLERANCE COLLAPSE 0.0001
CONVERGENCE TOLERANCE DISPLACEMENT 0.001
END
PERFORM PUSHOVER ANALYSIS                       1                (2f1)P

f1P

Displacement

June 22-25, 2011                   GTSUG, 2011, Delray Beach,FL                15
Simple Basic Behavior Examples
Simply-supported beam under axial load, imperfect geometry

Push-over Analysis Procedure

UNITS KIPS
21 FORCE X -1000.0
NONLINEAR EFFECTS
GEOMETRY MEMBERS EXISTING

PUSHOVER ANALYSIS DATA                     3
MAX NUMBER OF LOAD INCR 200
MAX NUMBER OF TRIALS 20
MAX NUMBER OF CYCLES 100
CONVERGENCE RATE     0.8
CONVERGENCE TOLERANCE COLLAPSE 0.0001
CONVERGENCE TOLERANCE DISPLACEMENT 0.001                               (3f1)P
END
1                    (2f1)P
PERFORM PUSHOVER ANALYSIS
f1P

Displacement

June 22-25, 2011                   GTSUG, 2011, Delray Beach,FL                         16
Simple Basic Behavior Examples
Simply-supported beam under axial load, imperfect geometry

Push-over Analysis Procedure

UNITS KIPS
21 FORCE X -1000.0
NONLINEAR EFFECTS
GEOMETRY MEMBERS EXISTING

PUSHOVER ANALYSIS DATA
3
MAX NUMBER OF LOAD INCR 200
4
MAX NUMBER OF TRIALS 20
MAX NUMBER OF CYCLES 100
CONVERGENCE RATE     0.8  \$ r
CONVERGENCE TOLERANCE COLLAPSE 0.0001
CONVERGENCE TOLERANCE DISPLACEMENT 0.001
END                                                                      (3f1)P
(2f1 + rf1)P
PERFORM PUSHOVER ANALYSIS
1                 (2f1)P

f1P

Displacement

June 22-25, 2011                    GTSUG, 2011, Delray Beach,FL                                    17
Simple Basic Behavior Examples
Shallow truss arch: snap-through behavior

P

u

L
3 in
3-u
L’
θ

2 @ 100 in

E = 29,000 ksi
Plane Truss: Ax = 1.0 in2

June 22-25, 2011                  GTSUG, 2011, Delray Beach,FL                18
Simple Basic Behavior Examples
Shallow truss arch: snap-through behavior

June 22-25, 2011              GTSUG, 2011, Delray Beach,FL     19
Simple Basic Behavior Examples
Shallow arch toggle: SBHQ6 model, snap-through behavior

P
Θz = 0
A

0.36 67 in
A

2 @ 1 2.94 3 in                Fixed (typ)

E = 1 .030 000 0E+07 lb s/in 2
ν = 0.0
Y

0.24 3 in
0.75 3 in
X
Section A-A

SBHQ6 Arch Leg, 20 x 4

June 22-25, 2011        GTSUG, 2011, Delray Beach,FL                           20
Simple Basic Behavior Examples
Shallow arch toggle: SBHQ6 model, snap-through behavior

Note: Pbuck = 152.4 lbs (linear buckling load)

June 22-25, 2011        GTSUG, 2011, Delray Beach,FL                                            21
Simple Basic Behavior Examples
Slender cantilever shear wall under axial load -- in-plane SBHQ plate behavior
P

0.01 kips

Mesh =   2X50
Material =    concrete
POISSON   = 0.0                               2 ft
Thickness    = 4 in
100 ft

June 22-25, 2011                   GTSUG, 2011, Delray Beach,FL             22
Simple Basic Behavior Examples
Slender cantilever shear wall under axial load -- in-plane SBH plate behavior

Pbuck (FE) = 41.95 kips

(Pe (SF) = 28.42 kips)

June 22-25, 2011         GTSUG, 2011, Delray Beach,FL                     23
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                      24
The P-δ Question

June 22-25, 2011      GTSUG, 2011, Delray Beach,FL   25
The P-δ Question

Mtot = M0 + Pδmid

June 22-25, 2011      GTSUG, 2011, Delray Beach,FL   26

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