Dynamic Analysis of Nuclear
Containments Using Shear
Deformation Shell
Dr. Mukti L. Das
Seattle, Washington
November 13-16, 2012
Plates And Shell Theories
To idealize a structure as a mathematical model, there is a need
for a structural element that has a small third dimension
compared to other two dimensions.
This idealization resulted to various plates/shell theories that
approximate equations of three dimensional quantum mechanics.
Two commonly used theories are, a) Kirchhoff-Love theory and
b) Mindlin - Reissner theory
In this presentation, all plates/shell theory will be referred as
“Shell Theory”.
Kirchhoff – Love Classical Shell Theory
This theory is an extension of Euler – Bernoulli beam theory. The following
assumptions are made in this theory:
• Straight lines initially normal to the mid-surface remain straight and normal
after deformation
• Thickness of shell remain unchanged during the deformation process
Mindlin – Reissner Moderately Thick Shell
Theory
This theory is based on following assumptions:
•
Straight lines initially normal to the mid-surface remain straight but may
not remain normal after deformation
•
Thickness of shell remain unchanged during the deformation process
Software Used
Kirchhoff – Love:
GT STRUDL (SBHQ6)
Mindlin – Reissner:
GT STRUDL (SBMITC, IPSQQ); ANSYS (SHELL43), STAAD (SHELL)
Experiment with a 20′X20′ Fixed-Fixed Plate
Deflection at Plate Center
E= 3,605.0 ksi
Poisson= 0.3
Uniform load = 1.0 ksf
7
6
Deflection (in)
5
Timoshenko
4
GTStrudl (SBHQ6)
3
GTStrudl (SBMITC)
2
STAAD
1
0
0.00
ANSYS
1.00
2.00
3.00
4.00
Thickness of Slab (ft)
5.00
6.00
Experiment with a 20′X20′ Fixed-Fixed Plate (cont’d)
Moment (kip-ft/ft)
Moment at Plate Center
10.5
10.4
10.3
10.2
10.1
10.0
9.9
9.8
9.7
9.6
9.5
9.4
9.3
9.2
9.1
0.00
Timoshenko
GTStrudl (SBHQ6)
GTStrudl (SBMITC)
STAAD
ANSYS
1.00
2.00
3.00
4.00
Thickness of Slab (ft)
5.00
6.00
Experiment with a Benchmark Reference
Cylinder
The article, “Consideration of Shear Deformation in the Analysis of
Unsymmetrical Bending of Moderately Thick Shell of Revolution”
published in the Transaction of 3rd SMiRT Conference, September 1975,
is adopted as an experimental benchmark.
Experiment with a Benchmark Reference
Cylinder (Cont’d)
The reference used a cylinder with the following data to demonstrate the
theory that was developed in the reference.
Diameter = 4 m
Height = 8 m
Internal Pressure = 1.0 Kg/cm2
E = 2.1 x 105 Kg/cm2
n
= 0.2
Experiment with a Benchmark Reference
Cylinder (Cont’d)
Fixed End Moment
Fixed - Fixed Condition
2400
2200
Moment (kg-cm/cm)
2000
SBHQ6
1800
1600
Benchmark
Claasical
1400
1200
Benchmark Shear
Deformation
1000
800
SBMITC
600
SBHQ6
400
200
0
1/5
1/6
1/7
1/8
1/9
1/10
1/11
1/12
1/13
Ratio t/R
1/14
1/15
1/16
1/17
1/18
1/19
1/20
Experiment with a Benchmark Reference
Cylinder (Cont’d)
Fixed End Moment
Moment (kg-cm/cm)
Fixed - Free Condition
2400
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0
SBHQ6
Benchmark
Classical
Benchmark Shear
Deformation
SBMITC
1/5
1/6
1/7
1/8
1/9
1/10
1/11
1/12
1/13
1/14
Ratio t/R
1/15
1/16
1/17
1/18
1/19
1/20
Experiment with a Containment
Major Design Parameters for Typical Nuclear Plants
Typical Power Plant
Model in Study
Diameter of Cylinder = 100′ – 130′
147′
Thickness of Cylinder = 3′ 6″ – 3′ 9″
3′ 9″
Thickness of Dome
= 2′ 6″ – 3′ 6″
3′ 3″
Thickness of Slab
= 8′ 6″ – 10′ 6″
3′ 3″ to 26′ 3″
Height of Cylinder
= 100 ′ – 169′
137′ 6″
Soil Class
= Sand – Hard rock
Accidental Pressure
= 60 psi – 200 psi
Loose sand ( Ks=48 k/ft3 )
143 psi
Experiment with a Containment (Cont’d)
A Typical Containment Model for this Study
Geometry:
Slab Diameter =48.25 m
Cylinder Diameter =45.25 m
Cylinder Height =39.40m
Total Height =59.00 m
Cylinder Thickness = 1.2 m (Constant)
Dome Thickness =1.0 m (Constant)
Base Mat Thickness = 1m, 2m, 4m, 8m & 12m (One
Particular Thickness at a time)
Support:
Soil Supported, Modeled as Winkler Spring
Loading:
1) Self Weight
2) Patch Load On Base Mat: 1379.46 kN/m2
(21.3mx21.3m)
3) Accidental Internal Pressure: 1000 kN/m2
4) Wind Load of 7 kN/m2 (141 km/h)
Experiment with a Containment (Cont’d)
Patch Load on the Base Mat
Patch Load:
1379.46 kN/m² on
21.34m X 21.34m
Experiment with a Containment (Cont’d)
Mid Point Deflection of Base Mat due to Patch Load
Slab Center Deflection Due to Patch Load on Mat
25
Deflection (cm)
20
15
SBHQ6
IPBQQ
STAAD
10
SBMITC
5
0
0
2
4
6
8
Slab Thickness (m)
10
12
14
Experiment with a Containment (Cont’d)
Moment About X-Axis on a Mid Point Element of Base Mat due to Patch
Load
Slab Center Deflection Due to Patch Load on Mat
25
Deflection (cm)
20
15
SBHQ6
IPBQQ
STAAD
10
SBMITC
5
X
0
0
2
4
6
8
Slab Thickness (m)
10
12
14
Experiment with a Containment (Cont’d)
Moment about X-Axis at Elv 6.47 m due to Patch Load
Moment about X-Axis at Elv. 6.47m Due to Patch Load
3500
Moment (kN-m/m)
3000
2500
SBHQ6
2000
IPBQQ
1500
SBMITC
1000
500
0
0
2
4
6
8
Slab Thickness (m)
X
10
12
14
Experiment with a Containment (Cont’d)
Deformed Shaped due to Accidental Internal Pressure
Experiment with a Containment (Cont’d)
Mid Point Deflection of Base Mat due to Accidental Internal Pressure
Slab Center Deflection Due to Accidental Pressure
18
16
Deflection (cm)
14
12
SBHQ6
10
SBMITC
8
STAAD
6
4
2
0
0
2
4
6
8
Slab Thickness (m)
10
12
14
Experiment with a Containment (Cont’d)
Moment About X-Axis on a Mid Point Element of Base Mat due to
Accidental Internal Pressure
Moment about X-Axis at Center of Slab Due to Accidental Pressure
100000
90000
Moment (kN-m/m)
80000
70000
60000
SBHQ6
50000
IPBQQ
40000
STAAD
30000
SBMITC
20000
10000
X
0
0
2
4
6
8
Slab Thickness (m)
10
12
14
Experiment with a Containment (Cont’d)
Moment about X-Axis at Elv 6.47 m due to Accidental Pressure
Moment about X-Axis at Elv. 6.47m Due to Accidental Pressure
14000
Moment (kN-m/m)
12000
10000
8000
SBHQ6
IPBQQ
6000
SBMITC
4000
2000
0
X
0
2
4
6
8
Slab Thickness (m)
10
12
14
Experiment with a Containment (Cont’d)
Moments about X-Axis at Elv 30.1 m And 52.55 m due to Accidental
Pressure
Moment (Kn-m/m)
Moment about X-Axis at Elev. 52.55 m Due to
Accidental Pressure
160
140
120
100
80
60
40
20
0
SBHQ6
IPBQQ (Base Only)
SBMITC
0
5
10
Slab Thickness (m)
15
X
Moment (kN-m/m)
Moment about X-Axis at Elev. 30.11 m Due to
Accidental Pressure
250
200
SBHQ6
150
IPBQQ (Base Only)
100
SBMITC
50
0
0
5
10
Slab Thickness (m)
15
Experiment with a Containment (Cont’d)
Moment at Elev. 63.17m Due to Accidental Internal Pressure
Moment (kN-m/m)
Moment about X-Axis at Elev. 63.17 m Due to
Accidental Pressure
90
80
70
60
50
40
30
20
10
0
SBHQ6
IPBQQ (Base Only)
SBMITC
0
5
10
Slab Thickness (m)
X
15
Experiment with a Containment (Cont’d)
Moment about Y-Axis at Location “A” on Base Mat due to Wind Load
Moment abot Y-Axis at Location A
3500
Wind Direction
Y
Moment (kN-m/m)
3000
2500
2000
SBHQ6
IPBQQ
SBMITC
1500
1000
500
0
Location A
0
5
10
Slab Thickness (m)
15
Eigenvalue Analysis of 10′ Diameter
Steel Plate With Fixed Edge
First Mode
450
400
350
Frequency (Hz)
300
250
200
150
100
SBHQ6
SBMITC
STAAD
50
0
0
1
2
3
Plate Thickness (ft)
4
5
6
Eigenvalue Analysis of 10′ Diameter
Steel Plate With Fixed Edge
20th Mode of 10' Diameter Steel Plate with Fixed
Edge
1600
Frequency (Hz)
1400
1200
1000
800
SBHQ6
600
SBMITC
400
STAAD
200
0
0
1
2
3
4
Plate Thickness (ft)
5
6
Eigenvalue Analysis of Containment
With Fixed Base
First Mode Frequency
First Mode Mass Participation
Dome:
1.0 m
Cylinder: 1.5 m
Mat Slab: 4.0 m
SBHQ6: 4.8 Hz
SBMITC: 4.8 Hz
STAAD: 4.8 Hz
SBHQ6: 66.1 %
SBMITS: 60.7 %
STAAD: 65.5 %
Dome:
2.0 m
Cylinder: 2.0 m
Mat Slab: 4.0 m
SBHQ6: 4.3 Hz
SBMITC: 4.3 Hz
STAAD: 4.3 HZ
SBHQ6: 71.2 %
SBMITC: 62.5 %
STAAD: 69.4 %
Dome:
4.0 m
Cylinder: 4.0 m
Mat Slab: 4.0 m
SBHQ6: 4.3 Hz
SBMITC: 4.3 Hz
STAAD: 4.3 Hz
SBHQ6: 70.7 %
SBMITC: 61.4 %
STAAD: 69.4 %
Dome:
1.00 m
Cylinder: 1.50 m
Mat Slab: 12.0 m
SBHQ6: 4.8 Hz
SBMITC: 4.8 Hz
STAAD: 4.8 Hz
SBHQ6: 66.1 %
SBMITC: 60.7 %
STAAD: 65.5 %