Dr. Mukti L. Das Seattle, Washington November 13-16, 2012 Dynamic Analysis of Nuclear Containments Using Shear Deformation Shell

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

Experiment with a 20′X20′ Fixed-Fixed Plate (cont’d) Moment at Plate Center

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 3 rd SMiRT Conference, September 1975, is adopted as an experimental benchmark.

Experiment with a Benchmark Reference Cylinder (Cont’d) Diameter = 4 m Height = 8 m Internal Pressure = 1.0 Kg/cm 2 E = 2.1 x 105 Kg/cm2  = 0.2 The reference used a cylinder with the following data to demonstrate the theory that was developed in the reference.

Experiment with a Benchmark Reference Cylinder (Cont’d) Fixed End Moment

Experiment with a Containment Major Design Parameters for Typical Nuclear Plants 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 Loose sand ( Ks=48 k/ft 3 ) Accidental Pressure = 60 psi – 200 psi 143 psi Typical Power Plant Model in 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: kN/m 2 (21.3mx21.3m) 3) Accidental Internal Pressure: 1000 kN/m 2 4) Wind Load of 7 kN/m 2 (141 km/h) Experiment with a Containment (Cont’d) A Typical Containment Model for this Study

Experiment with a Containment (Cont’d) Experiment with a Containment (Cont’d) Patch Load on the Base Mat Patch Load: kN/m² on 21.34m X 21.34m

Experiment with a Containment (Cont’d) Experiment with a Containment (Cont’d) Mid Point Deflection of Base Mat due to Patch Load

Experiment with a Containment (Cont’d) Experiment with a Containment (Cont’d) Moment About X-Axis on a Mid Point Element of Base Mat due to Patch Load X

Experiment with a Containment (Cont’d) Experiment with a Containment (Cont’d) Moment about X-Axis at Elv 6.47 m due to Patch Load X

Experiment with a Containment (Cont’d) Experiment with a Containment (Cont’d) Deformed Shaped due to Accidental Internal Pressure

Experiment with a Containment (Cont’d) Experiment with a Containment (Cont’d) Mid Point Deflection of Base Mat due to Accidental Internal Pressure

Experiment with a Containment (Cont’d) Experiment with a Containment (Cont’d) Moment About X-Axis on a Mid Point Element of Base Mat due to Accidental Internal Pressure X

Experiment with a Containment (Cont’d) Experiment with a Containment (Cont’d) Moment about X-Axis at Elv 6.47 m due to Accidental Pressure X

Experiment with a Containment (Cont’d) Experiment with a Containment (Cont’d) Moments about X-Axis at Elv 30.1 m And m due to Accidental Pressure X

Experiment with a Containment (Cont’d) Experiment with a Containment (Cont’d) Moment at Elev m Due to Accidental Internal Pressure X

Experiment with a Containment (Cont’d) Experiment with a Containment (Cont’d) Moment about Y-Axis at Location “A” on Base Mat due to Wind Load Wind Direction Location A Y

Eigenvalue Analysis of 10′ Diameter Steel Plate With Fixed Edge

Eigenvalue Analysis of Containment With Fixed Base Dome: 1.0 m Cylinder: 1.5 m Mat Slab: 4.0 m Dome: 2.0 m Cylinder: 2.0 m Mat Slab: 4.0 m Dome: 4.0 m Cylinder: 4.0 m Mat Slab: 4.0 m Dome: 1.00 m Cylinder: 1.50 m Mat Slab: 12.0 m SBHQ6: 4.3 Hz SBMITC: 4.3 Hz STAAD: 4.3 Hz SBHQ6: 4.8 Hz SBMITC: 4.8 Hz STAAD: 4.8 Hz SBHQ6: 4.8 Hz SBMITC: 4.8 Hz STAAD: 4.8 Hz SBHQ6: 4.3 Hz SBMITC: 4.3 Hz STAAD: 4.3 HZ First Mode Mass Participation SBHQ6: 66.1 % SBMITS: 60.7 % STAAD: 65.5 % SBHQ6: 71.2 % SBMITC: 62.5 % STAAD: 69.4 % SBHQ6: 70.7 % SBMITC: 61.4 % STAAD: 69.4 % SBHQ6: 66.1 % SBMITC: 60.7 % STAAD: 65.5 % First Mode Frequency