Transcript Simply-supported beam under axial load, imperfect geometry

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!
•
Additional Examples
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
T
B
 (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 :
T
B
 (u ) (u) dV  P  0
T
BT (u )  BLT  BNL
(u )
 (u )  D{ L   NL }

1  ui u j  u1 u1 u2 u2 u3 u3   




  ij  
  


2  x j
xi  x j x j x j x j x j x j   


T
T
{
B

B
 L NL (u )}D{ L   NL }dV  P  0
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 :
T
T
T
T
{
B
D


B
D


B
(
u
)
D


B
L
NL (u ) D NL } dV  P
 L L L NL NL
 L  [ BL ]{u},
 NL  [GNL (u )]{u}
T
{[
B
 L DBL ]{u} 
T
[ BLT DGNL (u )]{u}  [ BNL
(u ) DBL ]{u} 
T
[ BNL
(u ) DGNL (u )]{u}} dV  P
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
2
2
2
2
2



 uy
 uy
 u y   uz  
ux
 u z 1 u x
 uz
x 
 y 2  z 2  
 y 2  z 2  
 
 


x
x
x
2  x
x
x   x   x  


2
2
0
T
{[
B
L
 DBL ]{u} 
P and M are coupled



Modified UAxial and U Transverse are uncoupled 
θ

 Torsion and U Transverse are uncoupled
T
[ BLT DGNL (u )]{u}  [ BNL
(u ) DBL ]{u} 
0
T
[ BNL
(u ) DGNL (u )]{u}} dV  P
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:
T
B
 ( L NL) ( L NL) dV  P  0
d
 B


 ( L  NL ) dV u  P, where d 
u
T
( L  NL )
 dB(TL  NL ) ( L  NL ) dV  B(TL  NL ) d ( L  NL ) dV  u  P



 K  Ku  u  P;
June 22-25, 2011
KT u  P
GTSUG, 2011, Delray Beach,FL
9
Overview of Basic Concepts
The Tangent Stiffness Matrix
P
B
T
KT = [Kσ + Ku]
σdV
b
a
Pi+1
2
1
u2=u1+u2
Pi
u1
ui
June 22-25, 2011
u2
u1=ui+u1
GTSUG, 2011, Delray Beach,FL
ui+1
u
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
Plane Frame:
June 22-25, 2011
E = 10,000 ksi
Ax = 55.68 in2,
Iz = 100.00 in4
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
LOAD 1
JOINT LOADS
21 FORCE X -1000.0
$ Load P
NONLINEAR EFFECTS
GEOMETRY MEMBERS EXISTING
Load P
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 0.001
END
PERFORM PUSHOVER ANALYSIS
1
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
LOAD 1
JOINT LOADS
21 FORCE X -1000.0
NONLINEAR EFFECTS
GEOMETRY MEMBERS EXISTING
Load P
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 0.001
END
PERFORM PUSHOVER ANALYSIS
2
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
LOAD 1
JOINT LOADS
21 FORCE X -1000.0
NONLINEAR EFFECTS
GEOMETRY MEMBERS EXISTING
Load P
PUSHOVER ANALYSIS DATA
3
INCREMENTAL LOAD 1
MAX NUMBER OF LOAD INCR 200
MAX NUMBER OF TRIALS 20
MAX NUMBER OF CYCLES 100
2
LOADING RATE
0.005
$ f1
CONVERGENCE RATE
0.8
CONVERGENCE TOLERANCE COLLAPSE 0.0001
CONVERGENCE TOLERANCE DISPLACEMENT 0.001
END
1
PERFORM PUSHOVER ANALYSIS
(3f1)P
(2f1)P
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
LOAD 1
JOINT LOADS
21 FORCE X -1000.0
NONLINEAR EFFECTS
GEOMETRY MEMBERS EXISTING
Load P
PUSHOVER ANALYSIS DATA
3
INCREMENTAL LOAD 1
MAX NUMBER OF LOAD INCR 200
4
MAX NUMBER OF TRIALS 20
MAX NUMBER OF CYCLES 100
LOADING RATE
0.005
$ f1
2
CONVERGENCE RATE
0.8
$ r
CONVERGENCE TOLERANCE COLLAPSE 0.0001
CONVERGENCE TOLERANCE DISPLACEMENT 0.001
END
PERFORM PUSHOVER ANALYSIS
(3f1)P
1
(2f1 + rf1)P
(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
Thickness = 4 in
2 ft
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