Reliability Based Design Optimization of Primary Shield Structure Under Seismic Loading

RBDO OF PRIMARY SHIELD STRUCTURE
UNDER SEISMIC LOADING
S. K. Radha, A. Chakraborty
Virtual Integrated Analytics Solutions (VIAS)
K. C. Sener, A. H. Verma
Civil Engineering, Purdue University
October 30, 2018
1400 Broadfield Blvd. Suite 325, Houston TX 77084
Phone : +1 (832) 301-0881
www.viascorp.com
Contact Person:
Dr. A. Chakraborty – achakraborty@viascorp.com
SIMULIA Technology Day, Toronto

© 2018 Virtual Integrated Analytics Solutions Inc.© 2018 Virtual Integrated Analytics Solutions Inc.
Agenda
• VIAS Introduction
• Background
• Problem Statement
• Test Setup and Results
• FEA Model & Results
• Reliability based Design Optimization
• Optimization Results
▪ Plate Thickness
▪ Reliability
• Conclusions
• Future Work
2

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5

Introduction
• PSW provides support and protection to reactor
pressure vessel (RPV) and critical reactor internals of
PWR
• Consists of several structural members like bearing
walls, shear walls, slabs, etc.
• Seismic activity can damage critical structural/
shielding components
• Modular steel-plate composite (SC) PSW with
concrete infills increasingly used
• SC structures exhibits excellent structural
performance in terms of stiffness, strength and
ductility under seismic loading
RPV
PSW
Steel
Concrete
Steel-Plate
Composite PSW
Outer
Shield
Building
6

Introduction (Cont.)
• Typical SC structure consists of:
▪ Thick concrete walls (infill between plates)
▪ 3 layers of steel plates (interior and exterior
surfaces and 1 middle layer to provide
additional reinforcement)
▪ Intermediate web plates
• SC modules fabricated in production
environment and transported to construction
site
• Erected, welded and finally filled with concrete
on-site thereby reducing construction critical
path time
Exterior Plates
Interior Plates Web Plates
Concrete
Infill
Cavity
7

Background
• An experimental test setup of a 1/6th scale PSW
specimen and the seismic load-deformation
behavior tested
• 3D FEA model was developed and validated for
deterministic numerical study
Current work extends to reliability based design optimization (RBDO) study
8

Problem Statement
• Objective:
▪ Minimize T and t (minimizes inertia under
seismic and cost)
▪ Maximize Reliability (applied stress < yield)
• Design Variables:
▪ T: Steel plate thickness at openings
▪ t: Typical steel plate thickness
Steel plate thickness at openings
(T): Brown plates
Typical steel plate thickness (t):
Red, Green, Blue, and Black plates
Initial Design Dimensions
T= 0.17 in (4.32 mm)
t= 0.10 in (2.54 mm)
9

Test Setup
• Experimental test conducted in
Japan of 1/6th scale version of the
APWR PSW
• Polygonal SC structure with three
steel liner plates and web plates
• Free-standing test specimen
rigidly embedded in reinforced
concrete base
PSW Cross-section and Isometric view of PSW ( no concrete infill)
Horizontal Section Cut through
Specimen (with concrete infill)
Reinforced concrete base
(T)
(t)
10

Test Setup (Cont.)
• Test setup configured as a free
standing structure with top
connected to upper concrete
loading block and fixed base
• Lateral force applied on structure
hydraulically
• Quasi-static cyclic loading applied
• Subjected to three load-controlled
cycles in elastic range of response
• Final cycle until ultimate failure
11

Test Results
• Ultimate strength controlled by
combination of flexure, shear, and axial
forces in wall segments
• Local buckling and yielding occurred in
steel plates adjacent to openings
• Specimen showed substantial post yield
ductility
• Measured shear strains in inner, middle
and outer steel plates were relatively
equal
• Demonstrates that all three steel plates
contribute to system strength
12

FEA Model
• 3-D half model used since geometry and
loading are mirror symmetric
• All of the PSW steel plates are modeled using
4-node shell elements (S4R) and multi-axial
plasticity material model
• Concrete infill is modeled with continuum 3-D
8-node reduced integration (C3D8R) brick
elements and elastic fracture (brittle cracking)
material model
• Composite action between steel and concrete
is modeled with non-linear connector elements
• Monotonic and cyclic loading conditions are
simulated
Exploded view of part instances and mesh
13

FEA Model: Results
von Mises stress at applied loads of 1000,
2000, 3000 and 4000 kips on inner steel
plate
von Mises stress at applied loads of 1000, 2000, 3000
and 4000 kips on outer steel plate
von Mises stress at applied loads of 1000, 2000, 3000
and 4000 kips on middle steel plate
Red indicates yielding
(>67.3 ksi)
14

FEA Model: Results (Cont.)
• Analytical load-displacement response
behavior compares well with experimental
results
• Comparisons also indicate that cyclic load
response and behavior of specimen are similar
to monotonic analysis results
• Cyclic loading is not expected to have major
influence on results due to non-ductile shear
failure of specimen
• Hence only monotonic loading considered in
the RBDO study
Displacement gradient for monotonic loading
Comparison of Experimental and Analytical results
15

Requires Design Process Automation
DV2
Feasible
Region
DV1
(Sample points for DOE)
active constraint
inequality constraint
inactive constraint
Minimized
Objective Function
Contour
Reliable and
Robust Design
Deterministic Optimum
[High Failure Rate]
RBDO [Very Low Failure Rate]
Initial
Design from
DOE
(Approximation)
RBDO Process
16

RBDO Process (Cont.)
Requires Approximation of Response
17

Stochastic Analysis
• Maximum Von Mises stress (σvm) induced in steel
plate considered the most important criteria to
evaluate lateral load capacity
• T, t – design variables (to be within 20% of T = 0.17 in
and t = 0.1 in)
• T , t, σy, and u considered as normal random variables
with mean values and 10 % as their coefficient of
variation (COV)
• Performance function defined as g(X)= σvm (T, t, σy,
u) - σy
• Reaching the yield strength of steel (σy) is considered
as structural failure (i.e g(X) >0) (Failure Criteria)
• Target reliability 0.99
Normal Random Variable Mean
Thickness of steel plate at opening (T)
0.17 in
(4.32 mm)
Thickness of typical steel plate (t)
0.10 in
(2.54 mm)
Yield stress of Steel (𝛔 𝐲)
67 ksi
(462 MPa)
Applied Displacement (𝐮)
0.10 in
(2.54 mm)
18

Response Approximation
• Response Surface Method (RSM) is used for
optimization and reliability
• Response approximated as quadratic function that
provided good correlation (mean error< 5%)
60
62
64
66
60 62 64 66
Actual
Predicted
Actual vs. Predicted values of approximation.
25 FEA runs: For Approximation Function
10 FEA runs: For Error Computing
Typical FEA Runtime – 16hrs
19

Pareto Effects
• U, t has the most pronounced effect on von Mises
stress
• Higher order and correlation effects are small
0 10 20 30
σ_y
T
t
u
% effect on maximum von mises stress (Linear)
0 10 20 30
σ_y and T
σ_y and u
(σ_y)^2
T and u
t and u
u^2
% effect on maximum von Mises stress (2nd order
& correlated)
20

Optimization Workflow (Isight)
• RBDO workflow setup in Isight based on FEA model
response approximations
• Second Order Reliability Method (SORM) and Monte Carlo
Simulation (MCS) reliability techniques used
• Sequential Quadratic Programming (SQP) and Hooke-
Jeeves (HJ) direct search approaches used for optimization
Sample Size for Optimization –
DO : ~ 60
RBDO : ~ 100,000
21

Results: Optimized Plate Thickness
Reliability Method Optimization Method
T
(in)
t
(in)
SORM
SQP 0.16 0.12
HJ 0.16 0.12
MCS
SQP 0.14 0.12
HJ 0.14 0.12
RBDO:
Optimization Method
T
(in)
t
(in)
SQP 0.13 0.09
HJ 0.15 0.09
SQP 0.13 0.09
HJ 0.15 0.09
DO:
MCS results better
suited for non-
linear response
Original Values
T= 0.17 in
t= 0.10 in
22

Results: Reliability
Reliability
Method
Optimization Method DO RBDO
SORM
SQP 0.48 0.64
HJ 0.49 0.64
MCS
SQP 0.55 0.68
HJ 0.55 0.68
Original Model Reliability
SORM: 0.56
MCS: 0.62
23

Conclusions
• DO has lowered both thickness values from initial
• However, RBDO has lowered the thickness of steel plate at opening (T) by 17.64 % and
increased the thickness of typical steel plate (t) by 20 % from the original values
• RBDO achieved a reliability of 68 % and increased the reliability of the PSW structure
by 9.6 % and 23.6 % when compared to the original reliability and reliability from DO,
respectively.
24

Future Work
• Extension for a better RBDO of PSW which considers
• Additional failure criteria (concrete casting pressure, thermal loading, etc)
• Additional design and stochastic variables
25

Thank you
26

Reliability Based Design Optimization of Primary Shield Structure Under Seismic Loading

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