Standard Problem Exercise No. 3
Model 1: Tendon Behavior Model
April 13-14, 2011
Herman Graves
Lili Akin
Robert Dameron, PE
Patrick Chang, PE
Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation,
a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s
National Nuclear Security Administration under contract DE-AC04-94AL85000.
Standard Problem Exercise No. 3 Summary
• SPE No. 3 examines PCCV local effects and,
ultimately, developing pressure versus leakage
relationships
• First phase analysis focuses on:
2
- Effects of containment dilation on prestressing
force
- Slippage of prestressing and effects of force
- Steel-concrete interface
- Fracture mechanics behavior
- Scatter in data of prestressed concrete
properties
Model 1: Tendon Behavior Model
• Modeling assumptions, initial conditions, and
analysis results are presented for
1) Pressure only analysis
2) Pressure + temperature (saturated steam
condition) analysis
3
Model Geometry and Initial Conditions
4
Figure 1: Model 1 - Tendon Behavior Model, Representing Tendons H53 and H54, Elev. 6.579 m (Refer to Dwg. #
PCCV-QCON-04)
• Model consists of two hoop tendons, height of 225mm
(8-7/8”). Boundary conditions and pressure:
Model Geometry and Initial Conditions
• ABAQUS Standard FE program was used
• Model includes concrete, tendons, rebar (hoop
and shear reinforcement), and liner
• Concrete modeled with 8-node 3D solid elements;
Rebar modeled with embedded subelements;
Tendons with 2-node truss elements; Liner with
4-node shell elements, perfectly bonded to
concrete
5
Analytical Representation of Losses
1) Initial conditions applied to the tendons
2) FE Model‟s representation of angular friction
6
Figure 2: H53 Tendon Force Comparisons to Pretest (From NUPEC/NRC PCCV test at
SNL)
• For all participants to
begin their pressure
analysis from the
same basis, the
black line shows the
prescribed starting
point
Meridional Stress vs. Internal Pressure
• Relationship between the meridional stress, σm
and the internal pressure, p at level 6.579m is
prescribed by:
σm from dead load, prestress, internal pressure
= 7.02 – p*8.27MPa
(p in MPA, (+) compression, (-) tension)
(Equation developed by SPE Participant, Scanscot)
7
Additional Information About Tendon
Friction and Seating Losses
8
Material Modeling
• Tendon, rebar, and concrete material stress-strain
assumptions were implemented as tabulated in Appendix 1
of NUREG/CR-6810.
• Concrete simulated using ABAQUS concrete “Damaged
Plasticity”, smeared-cracking in tension (where cracking
occurs at element integration points) and a compressive
plasticity theory.
• Steel simulated using ABAQUS Standard Plasticity where
the stress-strain inputs consist of effective stress (Mises)
and effective strain. Inputs taken directly from SPE
Appendices.
9
Concrete Stress-Strain Curves
10
0.0
2.0
4.0
6.0
8.0
10.0
0.000 0.002 0.004 0.006 0.008 0.010
Str
ess (
ksi)
Strain
Concrete Compressive Strength
Figure 4: Concrete Compression Curve
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
-0.0005 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040
Str
ess (
ksi)
Strain
Concrete Tensile Strength
Figure 5: Concrete Tension Curve
Figure 6: Tendon Stress-Strain Curve
11
Figure 7: Liner Stress-Strain Curve
12
0
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
Stre
ss (
psi
)
Strain
Liner Stress vs. Strain
Liner
Figure 8: Rebar Stress-Strain Curve
13
Failure Criteria
• Relevant failure criteria for Model 1 is TENDON failure
• Rebar is not controlling since rebar has higher ductility
• Model 1 is not focused on liner tear/leakage
• Tendon Failure criteria taken as the Tendon System Elongation
(shown as strain) at Tendon rupture
• Different tests and different ways of measuring strain/elongation
• Reasonable consensus to use average of the Tendon System
Tests, or 3.8%
• One study suggested using 2% as a lower-bound criteria because
this is the limit-by-Specification (one tendon system test did show
a premature failure at under 2% due to anchor slippage)
• Tendon rupture at 2% is still considered to be a „possible‟ but not
„best-estimate‟ failure strain
14
Analysis Results
Required Output/Results for Model 1 • Description of Modeling Assumptions and
Phenomenological Models
• Description of Tendon Failure Criteria Used
• Pressure Milestones. Applied Pressure When:
- Concrete Hoop Stress (at 135° azimuth) Equals Zero
- Concrete Hoop Cracking Occurs (at 135° azimuth)
- Tendon A, and B Reach 1% Strain (at 135° azimuth)
- Tendon A, and B Reach 2% Strain (at 135° azimuth)
• Deformed Shape and Tendon Stress Distribution at P=0
(prestress applied); 1xPd; 1.5Pd; 2Pd; 3Pd; 3.3Pd; 3.4Pd;
Ultimate Pressure
• Description of Observations About Tendon Force as a
Function of Containment Dilation and Tendon Slippage
15
Model-1 ABAQUS Model
16
Tendon Layout
17
Anchorage of Tendon to Concrete
18
Rebar Layers Embedded in Concrete
19
Tendon Stress
20
Tendon Strain
21
Results by Pressure Milestones
Pressure Only Case
22
Conclusions from Model 1
• Tendon peak strains tend to be located at near
where strain is maximum after prestress anchor
set, i.e., azimuth 130-degrees
• But the “peak” moves around as the tendons
yield, reposition and slip relative to the concrete
• Circumferential slip of tendons relative to the
concrete is about 2 millimeters
• Using the contact surface method, such data as
shown are conveniently available
23
Animation of the Deformed Shapes at the
Required Pressure Milestones
24
Results of Radial Displacement vs.
Pressure at Different Azimuths
25
-20.0
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
180.0
0 0.3925 0.785 1.1775 1.57
Rad
ial D
isp
lace
me
nt
(mm
)
Pressure (MPa), Grid Division are multiples of Pd
Radial Displacement vs. Pressure
0°
90°
135°
270°
SOL #6
SOL #12
SFMT #6
Tendon Strains and Stresses vs. Pressure
26
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
-2.89E-15 0.3925 0.785 1.1775 1.57
Ten
do
n S
trai
n
Pressure (MPa), Grid Division are multiples of Pd
Tendon Strain vs. Pressure
Max Strain
Tendon A 135°
Tendon B 135°
Tendon Strains and Stresses vs. Pressure
27
600
800
1,000
1,200
1,400
1,600
1,800
2,000
-2.89E-15 0.3925 0.785 1.1775 1.57
Ten
do
n S
tre
ss (
MP
a)
Pressure (MPa), Grid Division are multiples of Pd
Tendon Stress vs. Pressure
Max Stress
Tendon A 135°
Tendon B 135°
Tendon Strains and Stresses vs. Pressure
28
124
126
128
130
132
134
136
138
-2.89E-15 0.3925 0.785 1.1775 1.57
Azi
mu
th (
de
gre
es)
Pressure (MPa), Grid Division are multiples of Pd
Location of Max Stress vs. Pressure
Liner Strains vs. Pressure
29
-0.001
0.001
0.003
0.005
0.007
0.009
0.011
0.013
0.015
-2.89E-15 0.3925 0.785 1.1775 1.57
Ho
op
Str
ain
Pressure (MPa), Grid Division are multiples of Pd
Liner Hoop Strain at 135° vs. Pressure
Circumferential Slip of Tendons
Relative to the Concrete
30
Circumferential Slip of Tendons
Relative to the Concrete
31
Pressure + Temperature Case
• Used „Saturated Steam‟ condition for a PCCV
• Pressure-temperature relationship applied to the inside face
• Used temperature distribution from ISP-48 thermal analysis
(through the thickness of the wall mid-height)
• Temperature triggers degradation of material properties
• Temperatures are not high enough to affect the steel, but
they are high enough to affect concrete
• The three layers of concrete elements through the
thickness of Model 1 were assigned slightly degraded
properties
• After prestressing and anchor set, an additional equilibrium
step is added where temperature is raised to 80°C, and
then the temperature and pressure are increased together
32
Pressure-Temperature Relationship
Applied to the Inside Face
33
0
20
40
60
80
100
120
140
160
180
200
0 50 100 150 200 250
Tem
pe
ratu
re (
C°)
Internal Pressure (MPa)
Change In Temperature on Inside Face
Temperature Variation Through Vessel
Wall (Ambient Temp = 21.1 °C
34
Concrete Degradation with Change in
Temperature
35
Results by Pressure Milestones
Pressure + Temperature Case
36
Temperature Case
37
-5.0
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
0 0.3925 0.785 1.1775 1.57
Rad
ial D
isp
lace
me
nt
(mm
)
Pressure (MPa), Grid Division are multiples of Pd
Radial Displacement vs. Pressure
0° 90°
135° 270°
SOL #6 SOL #12
Conclusions from
Pressure + Temperature Case
• No significantly different conclusions in terms of ultimate
limit state for PCCV for pressure + temperature
• Interesting phenomenon between 1Pd and 2Pd
- During this range, ovalized shape of “ring” changes from
“dimpled” at buttresses, to ovalized outward at the buttresses.
At larger pressures, shape of ring returns to similar pattern as
for pressure only analysis.
• Another difference, the tendon-slippage relative to the
concrete reaches 3.2mm, which is larger than the 1.8mm
observed for pressure only analysis
38
Temperature Case
39
0.003
0.005
0.007
0.009
0.011
0.013
0.015
0 0.3925 0.785 1.1775 1.57
Ten
do
n S
trai
n
Pressure (MPa), Grid Division are multiples of Pd
Tendon Strain vs. Pressure
Max Strain
Tendon A 135°
Tendon B 135°
Temperature Case
40
600
800
1,000
1,200
1,400
1,600
1,800
0 0.3925 0.785 1.1775 1.57
Ten
do
n S
tre
ss (
MP
a)
Pressure (MPa), Grid Division are multiples of Pd
Tendon Stress vs. Pressure
Max Stress
Tendon A 135°
Tendon B 135°
Temperature Case
41
124
126
128
130
132
134
136
138
0 0.3925 0.785 1.1775 1.57
Azi
mu
th (
de
gre
es)
Pressure (MPa), Grid Division are multiples of Pd
Location of Max Stress vs. Pressure
Temperature Case
42
-0.002
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0 0.3925 0.785 1.1775 1.57
Ho
op
Str
ain
Pressure (MPa), Grid Division are multiples of Pd
Liner Hoop Strain at 135° vs. Pressure
Temperature Case
43
Animation of the Deformed Shapes
Temperature Case
44
