ML16127A491
| ML16127A491 | |
| Person / Time | |
|---|---|
| Site: | Three Mile Island  |
| Issue date: | 04/11/2016 |
| From: | Exelon Generation Co |
| To: | Office of Nuclear Reactor Regulation |
| Shared Package | |
| ML16127A520 | List:
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| References | |
| TMI-16-035 | |
| Download: ML16127A491 (51) | |
Text
TMI-1 UFSAR CHAPTER 05 5C-i REV. 18, APRIL 2006
APPENDIX 5C
GILBERT ASSOCIATES, INC., REPORT
TO METROPOLITAN EDISON COMPANY
ON DESIGN OF LARGE OPENING REINFORCEMENTS FOR REACTOR BUILDING
FOR THREE MILE ISLAND NUCLEAR STATION UNIT 1
This Report Contains:
44 pages of text 8 tables 29 figures 2 drawings Appendix A (8 pages)
Appendix B (3 pages)
TMI-1 UFSAR CHAPTER 05 5C-ii REV. 18, APRIL 2006
DESIGN OF LARGE OPENING REINFORCEMENTS FOR REACTOR BUILDING METROPOLITAN EDISON COMPANY THREE MILE ISLAND NUCLEAR POWER STATION - UNIT NO. 1
BY GILBERT ASSOCIATES, INC.
TMI-1 UFSAR CHAPTER 05 5C-iii REV. 18, APRIL 2006 TABLE OF CONTENTS
ITEM TITLE
SUMMARY
1 DESIGN BASES
1.1 General
1.2 Design
Loads 1.3 Load Combinations
1.4 Material
Stress/Strain Criteria 1.5 Test Condition
1.6 Operating
Condition
2 GENERAL DESCRIPTION OF OPENING REINFORCEMENT
2.1 Introduction
2.2 Ring Reinforcement Splices
2.3 Normal
Shear at Edge of Opening
2.4 Prestressing
- 3. STRESS DISTRIBUTION AROUND A CIRCULAR HOLE IN A CIRCULAR CYLINDRICAL SHELL 3.1 Introduction 3.2 Finite-Element Method
3.3 Application
of Three-Dimensional Photoelasticity
4 ANALYSIS OF THE STRESS AROUND LARGE OPENINGS IN THE THREE MILE ISLAND REACTOR BUILDING 4.1 Verification of Finite-Element Method of Analysis
4.2 Stress
Analysis Under Operating and Accident Loads
4.2.1 Representation
of the Shell Around the Opening
4.2.2 Basic
Loading Conditions
4.2.3 Effect
of Creep and Shrinkage
5 VERIFICATION OF DESIGN CRITERIA
5.1 Basis
for Verification of Shell Loading Capacity Due to Primary Loads (Principal Stress-resultants and Principal Stress-couples) 5.2 Interaction Diagram
5.3 Reinforcing
Steel
5.4 Penetration
Barrel
5.5 Concrete
Shear
5.6 Shear
- Diagonal Tension
5.7 Shear
- Transfer Plates
5.8 Radial
Reinforcement
5.9 Summary
of Design and Conclusion
LIST OF REFERENCES
TABLES
FIGURES
DRAWINGS TMI-1 UFSAR CHAPTER 05 5C-iv REV. 18, APRIL 2006 LIST OF TABLES
NUMBER TITLE 5C-1 Fundamental Loads
5C-2 Stress Around Equipment Hatch Loading Condition No. 1 Dead Load
5C-3 Stress Around Equipment Hatch Loading Condition No. 2 Hoop & Verticle Prestress
5C-4 Stress Around Equipment Hatch Loading Condition No. 3 Operating Temperature (Winter) 5C-5 Stress Around Equipment Hatch Loading Condition No. 4 Accident Temperature (with Barrel Effect)
5C-6 Stress Around Equipment Hatch Loading Condition No. 5 55 Psig Internal Pressure 5C-7 Stress Around Equipment Hatch Loading Condition No. 6 Earthquake # 1
5C-8 Stress Around Equipment Hatch Loading Condition No. 7 Earthquake #2 TMI-1 UFSAR CHAPTER 05 5C-v REV. 18, APRIL 2006 LIST OF FIGURES
NUMBER TITLE 5C-1 Total Stress Concentration Factors For Given Loading Cases
5C-2 Stress Concentration Factor Variations For Different Bpo At Various Angles -
Capped Cylinder
5C-3 Stress Distribution Around Openings in Cylindrical Shells
5C-4 Grid For Finite Element Analysis of Stresses Around Equipment Hatch In RCB
5C-5 Layer Thickness And Designation
5C-6 Member Stresses Around Opening Edge (Vessel Subject To Internal Pressure)
5C-7 Surface Stresses Around Opening Edge (Vessel Subject To Internal Pressure)
5C-8 Hoop Stresses Along Longitudinal Axis (Vessel Subject To Internal Pressure)
5C-9 Axial Stresses Along Transverse Axis (Vessel Subject To Internal Pressure)
5C-10 Hoop Stress-Resultant N o Along Symmetry Axes (Test Problem)
5C-11 Effect of Tendon Curvature
5C-12 Nodal Forces Due to Curvature of Tendons In Neighborhood of Operating
5C-13 Normal Winter Operating Temperat ure Gradient-84" Thick Wall Near Equipment Opening 5C-14 Normal Winter Operating Temperat ure Gradient-42" Thick Wall Near Equipment Opening 5C-15 Normal Winter Operating Temperatur e Gradient-Transition Wall Near Equipment Opening 5C-16 Winter Accident Temperature Gr adient-84" Thick Wall Near Equipment Opening
5C-17 Winter Accident Temperature Gr adient-42" Thick Wall Near Equipment Opening
5C-18 Winter Accident Temperature Gr adient-Transition Wall Near Equipment Opening
5C-19 Interaction Diagram For Axial Compression/Tension And Bending
5C-20 Equipment Access Interaction Diagram -Elements 77 & 101
5C-21 Equipment Access Interaction Diagram - Elements 66 & 100
5C-22 Equipment Access Interaction Diagram - Elements 55 & 99 TMI-1 UFSAR CHAPTER 05 5C-vi REV. 18, APRIL 2006 5C-23 Equipment Access Interaction Diagram - Element 44
5C-24 Equipment Access Interaction Diagram - Element 73
5C-25 Equipment Access Interaction Diagram - Element 74
5C-26 Equipment Access Interaction Diagram - Elements 66 & 77
5C-27 Equipment Access Interaction Diagram - Elements 44 & 55
5C-28 Equipment Access Interaction Diagram - Elements 99, 100, & 101
5C-29 Sheer Transfer Plates TMI-1 UFSAR CHAPTER 05 5C-vii REV. 18, APRIL 2006 APPENDIX A
Effect of Concrete Creep and the Sustained Operating Stresses on Stress Distribution Around Openings in a Rapidly Pressurized Reinforced Concrete Vessel
APPENDIX B
Earthquake Analysis
TMI-1 UFSAR APPENDIX 5C 5C-1 REV. 18, APRIL 2006
SUMMARY
DESIGN BASES
The Reactor Building large openings, consisting of a 22 feet 4 inches diameter opening (equipment
access hatch) and a 10 feet 0 inches diameter opening (personnel lock), are designed generally in
accordance with the criteria as detailed in The Three Mile Island Nuclear Station Unit 1 Safety
Analysis Report. The steel reinforcements used in the vicinity of the openings have a minimum yield stress of 40,000 psi.
GENERAL DESCRIPTION
The vertical and horizontal prestressing tendons are draped around the holes and are continuous (i.e., no tendons are terminated at the hole). Circular rebar rings are located around the hole as
principal reinforcement. Horizontal and verical rebars are provided to the extent required by calculated radial (to the hole) stresses. Normal shear forces, due to pressure on the hatch, at the
interface between penetration barrel and concrete are resisted by steel shear plates.
STRESS DISTRIBUTION AROUND A CIRCULAR HOLE IN A CIRCULAR CYLINDRICAL SHELL A survey was made of the available theoret ical solutions and experimental techniques for determining stress distributions around circular holes in a shell structure. This survey indicated that an available programmed finite-element solution was most practical for this application.
ANALYSIS OF STRESSES AROUND LARGE OPENINGS
A. Method of Analysis The adequacy of the design of the openings was verified by the use of Computer Program FELAP (Reference 12) developed at the Franklin Institute Research Laboratories (FIRL).
This program provides for the representation of the shell by flat rectangular panels with
multiple layers and is used on the assumption that the perturbation on the shell introduced
by the presence of the opening is local. The basis for the derivation of this program is
described in Reference 13.
B. Verification of Method Accuracy In order to evaluate the accuracy of the solution method, a test problem was solved to develop stresses which could be directly compared with other theoretical solutions and
experimental results. Very satisfactory results were obtained as evidenced by the data presented in Section 4.1. This study also verified the adequacy of the grids used on the
analysis of the openings for the Three Mile Island - Unit No. 1 Reactor Building.
C. Basis for Analytical Model For purposes of the analysis, the shell was idealized by representing (1) the liner as an isotropic steel layer, (2) the horizontal or vertic al reinforcement as an orthotropic layer with no Poisson's ratio effect and no shear stiffness, and (3) the circular ring reinforcement with
zero shear stiffness. A more complete description of the model idealization is contained in
Section 4.2.1.
TMI-1 UFSAR APPENDIX 5C 5C-2 REV. 18, APRIL 2006 D. Load Combinations
The basic loads, including dead, internal pressure, earthquake, prestress, and thermal (operating and accident) loads, were combined in accordance with the basic criteria and
more fully tabulated as shown in Table 1. All loading conditions are more fully tabulated as
shown in Table 1. All loading conditions are more fully described in Section 4.2.2.
E. Creep and Shrinkage Effects
The effect of load redistribution due to concrete shrinkage and creep was investigated as described in Section 4.2.3 and found to be negligible.
F. Summary of Results
The stress-resultants and stress-couples for panels which were of interest in evaluating the adequacy of the design are summarized in Tables 2 through 8. The correctness, of the
values computed by the finite-element method was verified in several ways, as described in
Section 4.1 (c).
VERIFICATION OF REINFORCEMENT ADEQUACY
A. General Method
Principal stress-resultants and stress-couples were computed for all panels. Interaction diagrams were prepared based upon procedures for ultimate strength design of ACI 318-63.
Interaction diagrams for those panels found to be significant
are included as Figures 20 through 28, indicating the stress state for all pertinent loading combinations. The interaction diagrams show that sufficient reinforcement has been provided to carry all loads, including the full thermal stress-resultants and stress-couples.
B. Additional Considerations
Additional studies were performed to evaluate the acceptability of the penetration barrel and reinforcement for normal shears, which are described in Sections 5.4 and 5.5.
C. Design Drawings
Design drawings providing details of the opening reinforcement are included at the end of this report.
TMI-1 UFSAR APPENDIX 5C 5C-3 REV. 18, APRIL 2006 1 DESIGN BASES
1.1 General
The large openings in the Reactor Building are designed generally in accordance with the criteria as detailed in the Safety Analysis R eport, and as more completely detailed hereafter.
The large openings consist of the opening for the equipment access hatch, with a diameter
of 22 feet 4 inches, and the opening for the personnel lock, with a diameter of 10 feet 0
inches Although these criteria and the analytical and design methods described hereafter
apply to both openings, detailed results are provided only for the larger opening.
1.2 Design
Loads
The following loads were considered in the structural design of the Containment Vessel:
- a. Internal Pressure-Design Pressure is 55 psig
- b. Test Pressure - 63.3 psig (1.15 times design pressure)
- c. Live Loads
- d. External pressure - 2.5 psig
- e. Wind Load
- f. Thermal Loads
- 1. Accident
- 2. Operating
- 3. Test
- g. Seismic Ground Accelerations
- h. Dead Load
- i. Prestressing Load
- j. Tornado Load
The thermal loads on the Reactor Building and their variation with time, are developed on the basis of the blow-down transients as detailed in the Safety Analysis Report. The
openings, as a portion of a Class I structure, are designed on the basis of a horizontal
ground acceleration of 0.06g as the design earthquake, and of 0.12g as the maximum
hypothetical earthquake. The vertical component is taken as 2/3 of the horizontal
component.
The equipment access hatch consists of a single door located outboard of the Reactor Building shell with a personnel lock inset in the door. The barrel (penetration sleeve) for the
hatch is consequently subjected to the accident pressure. The isolated personnel lock
consists of double doors, one located inboard and the other outboard of the Reactor
Building shell. The design considers the consequence of both doors being closed during the
accident as well as the extremely remote possibility that either one of the doors is opened
during the accident.
1.3 Load Combinations
The design is based upon limiting load factors which are used as the ratio by which accident, earthquake, and wind loads are multiplied for design purposes to ensure that the TMI-1 UFSAR APPENDIX 5C 5C-4 REV. 18, APRIL 2006 load deformation behavior is essentially elastic. The loads utilized to determine the required limiting capacity of any structural element are computed as follows:
- a. C=0.95D + 1.5P + 1.OT
- b. C=0.95D + 1.25P + 1.OT' + 1.25E
- c. C=0.95D + 1.OP + 1.OT
+ 1.OE' d. C=0.95D + 1.OW t + 1.OP t Symbols used in the above equations are defined as follows:
C: Required load capacity of section D: Deal load of structure P: Accident pressure load - 55 psig T: Thermal loads based upon temperature transient associated with 1.5 times accident pressure. T': Thermal loads based upon temperature transient associated with 1.25 times accident pressure.
T: Thermal loads based upon temperature transient associated with accident pressure. E: Seismic load based upon 0.06g ground acceleration.
E': Seismic load based upon 0.12g ground acceleration.
W t: Wind loads based on 300 mph tornado.
P t: Pressure load based on an internal pressure of 3 psig difference between inside and outside of the Reactor Building.
The acceleration response spectra and structural damping are as detailed in the Safety Analysis Report.
1.4 Material
Stress/Strain Criteria
- a. Concrete Reinforcement The deformed bars used for concrete reinforcement in the Reactor Building, are intermediate grade billet steel conforming to ASTM A615-68 Grade 40 with a
guaranteed minimum yield strength of 40,000 psi.
The design limit for tension members (i.e., the capacity required for factored loads) is based upon the yield stress of the reinforcing steel. No mild steel reinforcement is designed to experience strains beyond the yield point at the factored loads. The
load capacity so determined is reduced by a capacity reduction factor "f" which
provides for the possibility that small adverse variation in material strengths, workmanship, dimensions, and control, while individually within required tolerances and the limits of good practice, occasionally may combine to result in an
under-capacity. The coefficient "f" is 0.95 for tension, 0.90 for flexure, and 0.85 for
diagonal tension, bond and anchorage.
- b. Prestressing Tendons
The steel tendons for prestressing consist of 169 - 1/4 in. diameter wires using a BBRV anchorage system and high tensile, br ight, cold drawn, and stress-relieved steel wires conforming to ASTM A421-59T, Type BA, "Specifications for Uncoated TMI-1 UFSAR APPENDIX 5C 5C-5 REV. 18, APRIL 2006 Stress-Relieved Wire for Prestressed Conc rete" with a minimum ultimate tensile stress of 240,000 psi.
The steel tendons are stressed during the post-tensioning operation to a maximum of 80 percent of ultimate strength and locked-off for an initial stress of 70 percent of ultimate
strength. The maximum effective prestress is determined taking into consideration
allowances for the following losses, which are deducted from the transfer prestress:
- 1. Elastic shortening of concrete
- 2. Creep of concrete
- 3. Shrinkage of Concrete
- 4. Steel relaxation
- 5. Frictional loss due to intended or unintended curvature of the tendons
In no event does the effective prestress exc eed 60 percent of the ultimate strength of the prestressing steel, or 80 percent of the nominal yield point stress of the prestressing steel, whichever is smaller. The design is based upon the steel tendons not being stressed
beyond the yield point as defined by ACI 318-63 when subjected to the factored loads.
- c. Structural Concrete
The structural concrete will have a minimum compressive strength of 5,000 psi in 28 days. Under operating conditions the allowable concrete stresses are in accordance
with ACI 318-63 Part IV-A, "Structural Analysis and Proportioning of Members-Working Stress Design," and Part V, Chapter 26, "Prestressed Concrete."
The Reactor Building, including the large openings, is checked for the factored load combinations and compared with what is gener ally defined as the yield strength of the structure. For concrete, the yield strength is defined, except as described hereafter by the allowable stresses in ACI 318-63, Part IV-B, "Structural Analysis and Proportioning of
Members - Ultimate Strength Design." Concrete cracking is assumed when when the
principal tensile stress based upon all loads including thermal loads, exceeds 6f C or 424 psi.
- d. Liner
The liner was designed as participating with the concrete shell in carrying membrane forces (See 5.1). The stress limits established for the liner are consistent with those
limits for stress intensity (i.e., the difference between the algebraically largest
principal stress and the algebraically smallest principal stress at a given point)
defined in Section III of the ASME Boiler and Pressure Vessel Code and based upon
a working strength design consistent with that code.
The liner is carbon steel plate conforming to ASTM A283-67, grade C. This material has a minimum yield stress of 30,000 psi and a minimum ultimate tensile strength of 55,000 psi.
The liner is normally 3/8 inch thick in the cy lindrical portion of the structure. In the immediate vicinity of the large opening it is increased to a 1 1/8 inch thick reinforcing plate.
TMI-1 UFSAR APPENDIX 5C 5C-6 REV. 18, APRIL 2006 The barrel and reinforcing plate are carbon steel conforming to ASTM A516 Grade 60 Fire box Quality modified to ASTM A300. This mate rial has a minimum yield stress of 32,000 psi and a minimum ultimate tensile strength of 60,000 psi.
1.5 Test Condition
No specific stress or strain limits are established for the test condition. The factored load combinations previously described have been establis hed so as to ensure that the response due to design loads is essentially elastic. A check is made to ensure that no significant
permanent deformation of the structure occurs during the test. This means that following
the test there should be no visible permanent distortion of the liner and only small hairline
cracks should exist in the concrete.
1.6 Operating
Condition
The load combinations relevant to operating conditions are determined as reflected in Section 4.2.2. Allowable stresses are based upon those stipulated in ACI 318-63.
2 GENERAL DESCRIPTION OF OPENING REINFORCEMENT
2.1 Introduction
The normal flow of stresses in the hoop and meridional direction is obstructed by the large access holes in the Reactor Building. During normal operation of the plant, horizontal and
vertical prestress forces are the single largest stress contributor. These forces do not create
any difficulty in transferring horizontal and vertical compressive stresses around the
opening, but give the necessary reserve compressive stresses to counterbalance the
horizontal and vertical strains resulting from the accident pressure load. The horizontal and
vertical stresses in the reinforcing steel ar e very small during normal operation, but will theoretically increase up to almost yield stress due to the factored pressure load. These
hoop and meridional forces under pressure load will be transferred around the opening by
counter acting prestress forces and by hoop, meridional, and circular ring reinforcing steel.
2.2 Ring Reinforcement Splices
The circular ring reinforcing steel will be spliced by use of a Cadweld Rebar Splice. These splices are located at points of low rebar stresses, and no more than 1/3 of the ring steel will
be spliced in one section. This should eliminate any slip between Cadweld and rebar that
could occur at high rebar stresses.
Radial tensile stresses out from the center of the access opening are created due to the pressure load. Hoop and meridional reinforcement is provided to carry these stresses out from the opening and will be terminated where "pur e" membrane stresses exist in the wall.
2.3 Normal
Shear at Edge of Opening
The peripheral or normal shear reinforcement in the concrete around the Penetration Barrel is designed for the computed shear at a distance equal to or greater than, d/2 = 42.0 in.,
from the edge of the opening per ACI 318-63, or for twice the normal shear due to internal
pressure on the hatch, whichever is the larger of the two valves.
TMI-1 UFSAR APPENDIX 5C 5C-7 REV. 18, APRIL 2006
2.4 Prestressing
The maximum curvature for any horizontal or vertical tendon, around an opening, is at a radius of not less than 25.0 feet. When tendons are draped around penetrations in a
manner in which a radial force is exerted on the penetration, within the diameter of the
penetration, the spacing of the tendons are not less than 10 inches. The distance between
the penetration and the centerline of the first tendon for the equipment access and
personnel access penetrations is not less than 12 inches. The distance between tendons
when two layers are required is not less than 18 inches. The total angular change of the
tendons was laid out to keep frictional losses within a satisfactory margin, and to satisfy a
practical execution of the job in the field.
3 STRESS DISTRIBUTION AROUND A CIRCULAR HOLE IN A CIRCULAR CYLINDRICAL SHELL
3.1 Introduction
The first theoretical treatment of the problem presented by openings in thin shell structures is commonly attributed to Lur'e 1 , who obtained an approximate solution for the stress distribution around a very small circular cut-out in a thin circular cylindrical shell subjected to a homogeneous biaxial stress state [
x'[P. Lur'e derived the following expression for the edge stress:
Equation 3.1
[= ([x + [P) - 2([x + [P) cos2f
+ 3(1 - v
- 2) X d 2 [2[x + ([P - 3[x) cos2f] 16Rt
in which:
[x = meridional stress in the shell without the opening
[P = hoop stress in the shell without the opening f = angular coordinate (see Figure 2)
R = shell radius t = shell thickness d = diameter of cut-out = 2r
Equation (3.1) does not include bending effects, which can be quite important in thin shells.
Therefore, Lur'e's solution represented a relat ively minor improvement on the flat plate solution (Reference 2). In fact, for many years the design of reinforcement around shell openings was based on the classical stress concentration factors of flat plate theory. In this
area solutions for openings reinforced by means of a symmetrical circular doubler plate
were obtained by Sezawa and Kubo (Reference 3), Gurney (Reference 4), and Beskin (Reference 5). Solutions for the stress distribution around unreinforced and ring-reinforced
holes in flat plates can be found in Savin's (Refer ence 6) extensive treatment of the subject.
The plane stress approach is not satisfactory, however, for large openings. In this case large stress-couples may appear around the edge of the opening even when the shell elsewhere
is in a pure membrane state of stress. For tunately, very valuable results have recently TMI-1 UFSAR APPENDIX 5C 5C-8 REV. 18, APRIL 2006 become available. Withum (Reference 7) inv estigated the stress distribution in a cylinder weakened by a hole, subjected to torsion, by using a perturbation scheme. This technique
was extended by Kline 'et al'(Reference 8) to determine the stresses around a circular cut-out in a pressurized circular cylindrical vessel. This work, carried out at the General
Technology Corporation with the support of the Bureau of Ships of the U. S. Navy, was part
of a systematic theoretical investigation of two problems: (References 9 and 10).
- a. determination of the state of stress in the vicinity of a circular hole on a circular cylindrical shell subject to internal pressure.
- b. determination of the stress distribution in two normally intersecting cylindrical shells.
The solution of problem (a), for internal pressure as well as other practical loading conditions, has been presented by Naghdi and Eringen (reference 10). Lekkerkerker (Reference 11), extending Lur'e's approach, s olved the same problem for axial tension and torsion. He found excellent correlation bet ween his solution and experimentally determined stresses (using electric resistance strain gages) in a mild steel tube subject to torsion.
Stress concentration factors from References 8 to 11 are presentated in Figures 1 and 2.
The coefficients given may be directly applied to the calculation of maximum stresses at the
edge of the opening.
Unfortunately, no theoretical solution is available for reinforced openings or for non-isotropic shells (e.g., orthotropic reinforced concrete shells). The complexity of these problems is
such that they can only be dealt with by means of numerical or experimental methods.
Among the many possible approaches, the f inite-element method and the stress-freezing technique of three-dimensional photoelasticity have become especially attractive in recent years and are briefly evaluated in Sections 3.2 and 3.3.
3.2 Finite-Element Method
Steady progress in the finite-element approach has led to the possibility of determining the stress distribution around reinforced openings in shell structures with good accuracy. The
most important advantage of this method is its generality: reinforcing rings, variable shell
thickness, and material orthotropicity, for example, may be incorporated into the analysis
without difficulty. Unfortunately, the accuracy of the solution must still be evaluated by test:
For instance, results obtained using two different grids may provide an indication of
covergence. Alternatively, results obtained using a prescribed grid size and pattern may be
checked against some of the theoretical solutions of Section 3.1 or against experimental
values. The latter approach was followed in connection with the analysis of the stresses
around the openings for the Three Mile Island Reactor Building.
The designs of the reinforcement around the openings in the Three Mile Island Reactor Building were verified by using Computer Program FELAP (Reference 12), developed at
Franklin Institute Research Laboratories (FIRL). The solution is based on a representation
of the shell as an assembly of flat rectangular panels. In the first order shell theory
described by Zudans (Reference 13), which form ed the basis for the FIRL finite-element solution, the rotation about the normal to the shell middle surface is taken equal to zero.
Consequently, this leaves only five degrees-of-freedom associated with each nodal point. It
must be noted that the model with five degrees-of-freedom at a corner point, while "compatible" for plate problems, is unbalanced for shell problems in the third rotation (Reference 14). Although usually this unbalance does not affect the accuracy of the TMI-1 UFSAR APPENDIX 5C 5C-9 REV. 18, APRIL 2006 solution, it can lead to unrealistic results (Reference 14). Therefore, in applications to nuclear power plants, a verification of the results becomes desirable.
Connor and Brebbia (Reference 15) developed a sti ffness matrix for a thin shell element of rectangular plan and also noted that good results can be obtained with the finite-element
method using non-compatible displacement expansions, which do not include all the rigid
body displacements (Reference 15). Accor ding to Connor and Brebbia (Reference 15), it appears that curved elements lead to better results, for the same element size and shape, and are therefore more efficient than flat elements. However, comparison of results
obtained using two different types of curved triangular elements (Refences 16 and 17) with flat elements (References 12 and 14) for the test problem discussed in Section 4.1 did not
substantiate that belief (See Figure 10). In fact, all finite-element results were close to
Eringen, Naghdi, and Thiel's (Reference 9) solution, which can be considered, within the
limitations of shallow shell theory, an "exact" solution.
3.3 Applications
of Three-Dimensional Photoelasticity
Until very recently, experimental methods constituted the only feasible approach to the stress analysis of geometrically complicated reinf orced openings in shell structures. In this area, the stress-freezing technique of three-dimensional photoelasticity appears to be the
most suitable experimental method. Outstanding studies of stresses around reinforced
openings in pressure vessels were carried out by Levan (Reference 18), Taylor and Lind (Reference 19), and Takahashi and Mark (Reference 20). In the latter, comparison of the
photoelastic results with a finite-element analysis of the axisymmetric thick-walled reinforced concrete vessel showed very close agreement between the two solutions. Durelli, del Rio, Parks, and Feng (Reference 21) carried out an experimental evaluation of the stress around an opening in a thin shell by means of brittle coatings, electrical and mechanical strain
gages, micromenters, and photoelasticity with the objective of comparing the accuracy and
advantages of each technique.
Photoelasticity was concluded to be the most effective experimental approach in this type of problem (Reference 21). In the fabrication of the model Durelli 'et al' (Reference 21) used a
Hysol 4290 epoxy resin which was found to give poor performance in recent tests (Reference 22). Mark and Riera (Reference 22) believe that the use of the improved model materials will lead to a considerable reduction in the dispersion of photoelastic readings, which was large in the past (Reference 23), and which still is the most important argument
against this experimental approach. In spite of the difficulties associated with the model
material, the photoelasticity determined in Reference (21) shows good correlation with
Eringen and Naghdi's (Reference 9) theoretical solution. Stress concentration factors
determined photoelastically by Durelli 'e t al' and by Houghton and A. Rothwell (Reference 24) by means of electric resi stance strain gages mounted on circular cylindrical aluminum shells are given in Figure 3(c). Figure 3(c) also shows the stress concentration factors computed by Eringen and Naghdi (Reference 9).
The photoelastic approach presents several advantages (Reference 25): 1) full field observations give clear understanding of overall behavior and permits recognition of
unsuspected critical regions, 2) measurement s are made with very small effective gage lengths so that high gradients can be studied in small models, and 3) basic instrumentation
is simple and inexpensive. Traditionally, models have been machine finished, but improving casting techniques have already permitted the fabr ication of complicated models without any noticeable edge effect. This represents two important steps forward: 1) model fabrication TMI-1 UFSAR APPENDIX 5C 5C-10 REV. 18, APRIL 2006 cost can be drastically reduced and 2) the technique may now be applied to complicated models that cannot be readily machined.
Ducts for prestressing "tendons" have succe ssfully been incorporated into epoxy models by using nylon piano cords, which are set in the model pr ior to casting as if they were rebars in a conventional reinforced concrete element (Reference 26). After the epoxy has hardened
they can be easily pulled out, leaving a perfect duct without residual stresses around the
walls. The photoelastic method, therefore, offers the possibility of determining in one single
study the stresses around openings in prestressed vessels in "the large," as well as in "the
small." The latter, which includes stresses around curved tendons, corners, possible
non-linear distributions through the wall thickness, etc., may demand separate analyses
when a solution based on thin shell theory is used as the basis for design.
Further progress in theoretical or num erical analysis of stresses around shell openings, which must account for non-linear distributions through the shell thickness near the opening
and for other effects that cannot be predicted by thin-shell theory, can be fostered by
adequate experimental verification of the results, or by purely experimental studies that may help define the areas that require additional investigation. This is illustrated by a
photoelastic investigation of stresses around reinforced openings in plates due to
Lerchenthal (Reference 27), which indicates that the departure from a plane stress
distribution near the openings may be significant. For this purpose, the stress-freezing
technique may yet prove to be an invaluable tool.
TMI-1 UFSAR APPENDIX 5C 5C-11 REV. 18, APRIL 2006
- 4. ANALYSIS OF THE STRESSES AROUND LARGE OPENINGS IN THE THREE-MILE ISLAND UNIT NO. 1 REACTOR BUILDING
4.1 Verification
of Finite-Element Method of A nalysis as outlined in Section 3.1, a finite-element solution was chosen to determine stresses around large openings in the Reactor Building.
To evaluate the accuracy of the solution method (FIRL Program FELAP, as described in
Section 3.2), a test problem was solved with the same grid used in the finite-element
analysis of openings for the Three Mile Island Reactor Building. The grid is shown in Figure
- 4. In the finite-element analysis the following assumptions were made:
- a. The perturbation introduced by the presence of the opening on the shell state of stress is localized.
- b. According to (a.), stresses and displacem ents some distance away from the region surrounding the opening are not affected by the opening.
- c. A panel of rectangular plan, which is centered around the opening, is removed from the shell (Figure 4). The displacements corresponding to the shell without the
opening along the boundary lines of this panel constitute the boundary conditions for
the finite-element analysis. The analysis is correct if the panel boundaries are
sufficiently far from the opening so that (a.) and (b.) apply. This assumption can be
verified 'a posteriori' by comparing stresses along the boundary lines with those
existing in the shell without the opening.
- d. Each element can be idealized as a layered plate. Three typical elements, each with four layers, for representation of three regions of different thickness of the finite
element model, as shown in Figure 5.
- e. Because of symmetry, only one quadrant need be analyzed.
Since reliable experimental or theoretical result s for reinforced openings in shells similar to that under consideration were not available, it was decided to check the solution method
against the shell tested by Durelli 'et al' (Reference 21), for which other theoretical solutions
were also available. The example problem is defined by:
R = 430.00 in.
r = 85.70 in.
t = 18.04 in.
v = 0.30 p = 100.00 psi
The computed tangential membrane stresses around the opening edge are compared in Figure 6 with those given in Reference (9) and with the experimental stresses determined by Durelli 'et al' (Reference 21). Similarly, Figure 7 shows the tangential surface stresses
around the edge. Additional comparisons between the finite-element solution and the
experimental values are given in Figures 8 and
- 9. Finally, Figure 10 shows the variation along the symmetry axes of the stress resultant N P determined by different approaches, including another finite-element solution using triangular shell elements based on Prato's
work (Reference 16). The correlation of the FIRL finite-element solution with the other
results is good. It must be noted that the finite-element results are particularly close to the TMI-1 UFSAR APPENDIX 5C 5C-12 REV. 18, APRIL 2006 solution of Eringen 'et al' (Reference 9), which was regarded as the most accurate. It should also be pointed out that the correlation is good in spite of the fact that the panel boundaries
were not sufficiently removed from the opening, as revealed by Figure 9 which shows the existence of a small stress couple M P at s = 3.5 that had not been predicted by r
the model study. This result was verified by another finite-element analysis of the test problem using Prato's triangular shell elements (Reference 16).
It was pointed out in Section 3.1 that the stress concentration factors under loading conditions such as internal pressure, axial tension or torsion can be computed at different
locations in terms of the nondimensional parameter T (see Figure 2). As T 8 we approach
the plane stress solution and the convergence problem for the finite-element solution
disappears. It can be concluded, therefore, that satisfactory results in the test problem (T =
1.17) constitute adequate verification of the solution method for the large openings in the
Three Mile Island Reactor Building for which T = 62 (equipment hatch, based on typical shell
thickness).
4.2 Stress
Analysis Under Operating and Accident Loads
4.2.1 Representation
of the Shell Around the Opening
The finite-element grid used to solve the test problem (Section 4.1) was also employed to determine the stress distribution around the large openings in the Reactor Building. The
shell was idealized as follows:
- a. The liner was represented as an isotropic steel layer (E st = 30,000 ksi, U = 0.3).
Composite action was assumed in the determination of the stress resultants and
stress couples.
- b. The vertical and horizontal steel reinforcement was represented as a layer of an orthotropic material having no Poisson's ratio effect, no shear stiffness (G 12 = 8), and no meridional stiffness(E 1 = 8). This layer was located at the center of gravity of the meridional or hoop reinforcement.
- c. The ring steel rebars, providing additional reinforcement around the opening, were represented as a layer located at the center of gravity of the ring reinforcement. The shear stiffness of this layer was set to equal zero (G 12 = 8). In Computer Program FELAP the axes of orthotropicity are oriented in the hoop and meridional directions.
Consequently, since the axes of orthotropic ity of the ring reinforcement only coincide with those directions in the regions around the opening's vertical and horizontal axes, the following approximation was made:
at every panel type the hoop and meridional stiffnesses were directly proportioned to the projected steel area.
TMI-1 UFSAR APPENDIX 5C 5C-13 REV. 18, APRIL 2006
- d. The concrete layers were idealized as follows:
E 1 = 4000 ksi E 2 = 4000 ksi G 12 = 1740 ksi U 12 = 0.15
- e. The barrel (penetration sleeve) was represented on contributing, with 50 percent of its area, to the stiffness of the elements adjacent to the opening. In other words, a
stiffener of 3/8 inch thickness was included in the model along the periphery of the
opening.
Note that although the penetration barrel was incorporated into the finite-element model as explained above, it was not regar ded as contributing to the load-carrying capacity of the shell (See Section 5.2).
4.2.2 Basic
Loading Conditions
The stress distribution around the opening was determined for the loading conditions described in Section 1 and those loading combinations more completely described
hereafter. The specific loads are defined as follows:
- a. Dead Load
The stress distribution around the opening due to dead weight was calculated assuming a uniform meridional compression in the cylinder equal to the stress
resultant at the elevation of the opening axis (in the shell without the opening). The
weights of the equipment hatch and personnel lock were neglected. Since dead
weight stresses in the typical shell w all at the elevation of the equipment and personnel access are low (less than 100spsi), the above simplifications will not
significantly influence the final stresses in the pressurized vessel.
- b. Internal Pressure
Internal pressure was assumed to act on the interior of the shell as well as on the barrel (penetration sleeve) of the equipment access hatch. The internal pressure on
the hatch was assumed transferred to the shell by a uniform normal shear Q p around the opening. The effect of non-uniform distributions of Q p were analyzed separately.
- c. Earthquake
Seismic stresses around the openings were investigated for two directions of the horizontal component of motion:
- 1. Earthquake motion oriented in the direction normal to the openings.
- 2. Earthquake motion oriented at 900 with the direction normal to the openings.
Seismic loads were evaluated as described in Appendix B.
TMI-1 UFSAR APPENDIX 5C 5C-14 REV. 18, APRIL 2006
- d. Prestress
Prestressing loads are represented as two independent loading conditions by a uniform hoop and meridional compression in the shell (away from the opening) equal
to the stress resultant corresponding to initial and final prestress. Curved tendons
around the opening were considered in the analysis as line loads P = T/r where T is
the total prestressing force per tendon and r is the radius of curvature at the location
under consideration. These line loads were integrated within each panel and applied
as nodal forces on the shell, as shown in Figures 11 and 12.
- e. Thermal Loads
Operating Temperature
The steady-state temperature distribution in the reinforced area around the opening was obtained using "Heating 2" a computed code developed by Mr. R. D. Brunell of
Atomics International (Reference 28). This particular computer program is applicable for the solution of transient and/or steady state problems in one, two, or
three dimension cartesian or cy lindrical coordinate systems.
The steady state temperatures were determined using the formula:
M j K i (T i - T j) = 8 i=l
with convergence criterion
T j step - T j step A - 1 £ .00001 T j step A for all nodes.
The transient temperatures were determined using the formula:
M
j K i (T i - T j) = C j delta T j i=l delta T
to ensure stability of the transient solution the time step was chosen as:
delta T £ ( C j ) M j K i i=l
which was calculated for all nodes and the smallest value chosen.
The opening temperatures were determined for:
- 2. start-up in winter 3. shut-down in winter
- 4. accident condition in winter
The winter season was chosen because at this time the largest thermal gradients are present in the Reactor Building. The temperature on the external surface of the Reactor
Building was taken as 21.4F. This is the lowes t average monthly temperature ever recorded in the area by the United States Weather Bu reau. Lower temperatures will be experienced for shorter periods of time, however these only affect a small portion of the wall thickness, as proved by preliminary tests, and were not considered.
It was assumed that during start-up, the norm al operating temperature rose from 60 F to 110 F in one hour and for shut-down the reverse was used. The inside film coefficient was taken
as .010354 Btu/hr in 2 F for cases 1, 2, and 3 which is actually a modified film coefficient which takes into account the steel liner and the stagnant air gap between the liner and the
concrete:
h M = 1 X
STEEL X AIR 1
+
+ h i K STEEL K AIR
This does neglect any circumferential conduct ion, however, due to the small comparative thickness of the liner and air gap the effect will be negligible. This was also proven in
preliminary tests.
In case 4 the liner temperature was known therefore as a film coefficient of .040766 Btu/hr in 2 F was used which simply simulated the conduction through the air gap:
k AIR h M = X AIR The outside film coefficient in all cases was taken as .0069555 Btu/hr in 2 F.
The opening itself was broken into 61 nodes with adiabatic boundaries at the radial midpoint of the buttress and the radial midpoint of the plane wall. This was done because
considering symmetry, there is no heat transfer across these boundaries.
The following material values were used for the concrete:
P = .08102 lb/in.
3 C p = .210 Btu/lb F k = .06944 Btu/hr in F.
The input for the finite-element stress analysi s was prepared on the basis of the above results.
Summer thermal stresses are conservatively computed as equal to 27 percent of the peak winter stresses.
TMI-1 UFSAR APPENDIX 5C 5C-16 REV. 18, APRIL 2006
- f. Accident Temperature
The effect on the barrel (penetration) resulting from an accident temperature of 281F (net increase: 281 - 110 = 171F) was representated by internal pressure acting on the barrel.
The magnitude of this pressure was determ ined on the basis of a two dimensional analysis, which assumed the barrel (t = 0.75 in.) plus 0.55 in. of concrete were suddenly heated to
171 F. The equivalent pressure on the concrete was found to be equal to 183 psi.
Definitions for the foregoing variables are:
j K i = thermal conductivity between node j and i T i = temperature at node i T j = temperature at node J M = number of nodes adjacent to node j A = interation number
C j = compositance of node j T = time step X = thickness k = thermal conductivity
h M = modified film coefficient, inside h i = inside film coefficient h 8 = outside film coefficient P = density
C P = specific heat P 1 = radial distance between boundaries P 2 = radial distance opening covers
The temperature gradients obtained by t he aforemention computer program are presented as 6 figures (Figures 13 through 18). Figures 13, 14, and 15 refer to the
temperature gradients under winter operat ing temperature conditions at region near the opening, normal wall section, and the barrel-normal wall transition zone, respectively. Figures 16, 17, and 18 show the temperature gradients under winter accident temperature condition at region near the opening, normal wall section, and
the barrel-normal wall transition zone, respectively.
4.2.3 Effect
of Creep and Shrinkage
It appears that shrinkage of concrete can only introduce compressive stresses into the steel rebars. These stresses will largely disappear after the internal pressure in the shell takes
place, and need not be considered in the stress analysis of the opening.
The load redistribution due to concrete creep (i.e., the redistribution of Nf , NfP, NP , Mf, MP, etc.) is expected to be small. This conc lusion is sustained by a finite-element plane stress analysis, which indicates that the stress concentration factor for uniaxial compression
changes by only 5 percent when the hoop 'in-plane' stiffness is reduced by 100 percent. If the 'effective modulus' approach is used to determine the final stress distribution under
operating stresses, a final modulus equal to 65 percent of its initial value would lead to a
final ratio between vertical and hoop 'in-plane' stiffness equal to:
42 x 1600
= 67.3
= .950 41.875 x 1600 + .125 x 30,000 70.65 TMI-1 UFSAR APPENDIX 5C 5C-17 REV. 18, APRIL 2006 the initial ratio is:
42 x 4000
= 168
= .985 41.875 x 4000 + .125 x 30,000 170.8
Therefore, a change in the ratio between vertical and hoop 'in-plane' stiffnesses of about 4 percent is not expected to have any significant effect on the stress-resultant and
stress-couple distribution.
The effect of creep and shrinkage on prestress losses is taken into account as indicated in Section 5.8. Likewise, the transfer of load fr om concrete to steel in any given cross-section is considered in the verification of rebar stresses when applicable.
With reference to the discussion of Appendix A, it may be concluded that the load redistribution due to concrete creep in the unpressurized vessel will not affect in any
significant degree the load distribution in the structure under test or accident pressure.
- 5. VERIFICATION OF DESIGN CRITERIA
5.1 Basis
for Verification of Shell Loading Capacity Due to Primary Loads (Principal Stress-resultants and Principal Stress-couples)
The loading capacity at any point of the shell was verified according to the following procedure:
- a. The principal stress-resultants N 1 and N 2 were computed in terms of Nf, NP, and NfP.
- b. The principal stress-couples M 1 and M 2 were computed in terms of Mf, MP, and MfP.
- c. Considering that throughout the critical regions of the shell (both axes of symmetry and along the edge of the opening) the orientations of N 1 , N 2 , and M 1 , M 2 coincide with the orientation of the reinforcement and that in the rest of the shell they nearly coincide with each other and with the orientation of the steel rebars. Systems N 1 , M 1 , and N 2 , M 2 are treated independently. Since in the regions in which the directions of principal stress-resultants and stress-couples are not colinear or do not
coincide with the orientation of the steel rebars, stresses are low, the error
introduced by assuming them colinear will not affect the conclusion concerning the
load-carrying capacity of the shell. In the latter case, steel rebars not oriented in the
direction of principal stress-resultants or stress-couples were conservatively
neglected in the computation of the strength of the section. In other words, it is
proposed that forces in the 1-direction will not affect the strength of the section in the
2 direction and vice versa. This is in full agreement with a square failure criterion for
concrete in biaxial compression, which appears to be very conservative. Stresses in
rebars oriented along a principal stress direct ion obviously will not be affected by forces in the other principal direction.
TMI-1 UFSAR APPENDIX 5C 5C-18 REV. 18, APRIL 2006
- d. The computed ultimate capacity of any se ction of the shell satisfies the requirements of ACI 318-63, Sections 1600, 1700, 1800, and 1900. Only deformed bars as
defined in ACI 318-63, Sections 301 are used.
- e. Composite action between the shell and the liner is neglected in the computation of ultimate moments. The liner is regarded as carrying only its share of the principal stress-resultants.
Interaction diagrams were prepared as described in Section 5.2 for elements located along both symmetry axes and along a 45 degree line. Principal stress-resultants
and stress-couples corresponding to all critical load combinations are shown in
Figures 20 thru 28 on the interaction diagrams corresponding to elements 44, 55, 66, 77, 73, 74, 99, 100, and 101. The position of a point representing a stress state
within the diagram gives a clear indication of t he "local safety factor" at that location (i.e., at the center of the element). It should be pointed out that all points on the
interaction diagram are computed with respect to shell reference surface as shown in
Figures 20 thru 28. That is, under compression and bending, "N" and "M" were
transferred from the plastic centroid to the shell reference surface. Under tension, "N" was transferred from the center of gravity of the reinforcing steel to the shell
reference surface. It must be emphasized that even if a point representing a stress
state fell outside the diagram, that would not indicate a critical or nearly critical condition for two reasons:
- a. The interaction diagrams were determined on the basis of conservative assumptions and it is expected that the 'true' failure envelopes lie a certain
distance away from the computed envelopes.
- b. A point outside the interaction diagram would merely indicate local yielding of one or more rebars at that location, which would cause a load redistribution
towards less highly stressed regions. A point representing a stress state
contained within an interaction diagram indicates that stresses in all steel
rebars at that section are below yield stress, and that concrete stresses, where applicable, are below code allowable stresses.
5.2 Interaction
Diagram
- a. Axial Compression and Bending (See Figure 19)
- 1. Concentric Compression
P 8 = f [0.85 f C A C + A st f y],
A st = A s1 + A s2 + A s3
- 2. Simple Bending
a = A s1 f y + A s2 f s2 - A's3 f s3 0.85 f C b M 8 = [T 1 d 1 + T 2 d 2 + C s3 d'3 + C C (C - a)] 2 C d
TMI-1 UFSAR APPENDIX 5C 5C-19 REV. 18, APRIL 2006 M C M s1 + M C C C = 0.85 f C a b
- 3. Bending and Axial Compression
C b = d(87000) 87000 + f y a b = k 1 a.b P b = f [0.85 f C (a b - a) b]
M b = M 8 + P b d b
- 4. Determine P a and M a e a = 0.1 t; ACI 318-63, Section 1901 (a)
M a = P a e a By similar triangles, P a = P 8 M b e
a (P 8 - P b) + M b
- b. Axial Tension and Bending
- 1. Concentric Tension
P 8 = f A st f y Note: For purpose of clarity, only three steel layers
were included in the preceding equations.
5.3 Reinforcing
Steel
- a. The opening steel reinforcement will be as shown on Drawing Nos. D-421-035 and D-421-036.
- b. The amount of reinforcement, which includes regular hoop, meridional, and circular ring steel around the opening, equals or exceeds that shown to be required by
calculations. In no event is the liner assumed to contribute more than its yield
strength.
- c. The clear distance between parallel bars is not less than 2 times the maximum size of coarse aggregate, 2 times the bar diameter, nor less than 2 inches.
- d. Vertical shrinkage or temperature reinforcement is placed at the outside face of wall.
The minimum amount of such reinforcement on the outside concrete face wall is greater than 0.0015 of the gross cross-sectional area of concrete.
5.4 Penetration
Barrel
TMI-1 UFSAR APPENDIX 5C 5C-20 REV. 18, APRIL 2006 The portions of the Equipment Access Hatch and Personnel Lock extending beyond the concrete shell were designed and fabricated by Chicago Bridge and Iron Company in
accordance with the ASME Nuclear Vessels Code. The barrel of each of these penetrations
within the limit of the concrete thickness was investigated for the following loads:
- a. Meridional membrane stresses in the barrel due to internal pressure on the hatch.
- b. Hoop membrane stresses in the barrel due to the 'in-plane' deformation of the opening.
- c. Meridional bending stresses in the barrel caused by meridional shear transfer from the barrel to the concrete.
- d. Thermal Stresses.
The objective of this investigation was to verify that the stresses in the barrel are within allowable limits. Refer to ASME Nuclear Vessels Code, Article 4, Par. N-414.
It should be noted that the allowable stresses referred to are based on working
strength design.
5.5 Concrete
Shear
Splitting planes were hypothesized parallel to the surface of the shell through the various layers of concrete reinforcement and tendon conduit. The 'in-plane' shear stresses are
produced by radial forces produced by circular rebar rings and draped tendons. Sufficient
steel has been provided in the form of straight or hooked vertical and horizontal bars and
ties to develop the total shear stress across the hypothesized planes (Reference 29). The
shear stresses are conservatively assumed to be the summation of the loads resisted by the circular bars on the vertical axis due to the factored pressure load. That is to say, the
shearing force exerted across a plane through Layer 15 (see attached Drawing No.
D-421-036) is equal to the summation of rebar forces on Layers 15 and 16 on the vertical
axis. The maximum shear stress on the dowels due to the aforementioned load does not
exceed the yield stress of the dowels.
The dowels are anchored by mechanical anchorage (900 hooks) and/or sufficient bond development lengt h which is determined on the basis of Ultimate Strength Design provision of ACI 318-63. All rebars provided to resist the
aforementioned loads consist of A615 mater ial with a 40,000 psi minimum guaranteed yield strength.
5.6 Shear-Diagonal Tension
The ACI 318-63 recognizes the punching shear to be critical at a distance "d/2" out from the periphery for slabs and footings (See Sections 1207 and 1707). A beam type of diagonal tension failure is impossible due to the geometry and two directional stresses in the shell. We believe that a punching mode of failure is the type of
failure that should be and has been investigated. These shear stresses included in the
computer output take into account the effect of the pressure on the door plus the reinforced
area.
Two modes of shear transfer are considered. First, shear transfer through concrete without shear reinforcement. Second, disregarding the shear capacity of concrete, enough
reinforcing steel is provided to carry the normal shears by steel alone.
TMI-1 UFSAR APPENDIX 5C 5C-21 REV. 18, APRIL 2006 Ultimate peripheral or normal shear stress carried by concrete is computed by:
v u = Q d
v c = 4 f d (f C)1/2 (ACI 318-63, Section 1707) where:
Q = normal shear stress-resultant at the critical section
v u = nominal ultimate shear stress as a measure of diagonal tension
v c = allowable ultimate shear stress to be carried by concrete d = distance from extreme compression fiber to centroid of tension reinforcement f C = 28 days compressive strength of concrete f = 0.85
Shear reinforcement requirements
The ultimate shear capacity of the reinforcing steel alone is computed by:
v u = f [a sv f y] where:
A sv = cross-sectional area of reinforcing steel acting in tension across a potential diagonal tension crack
f y = yield strength of reinforcement f = 0.85 TMI-1 UFSAR APPENDIX 5C 5C-22 REV. 18, APRIL 2006
5.7 Shear
Transfer Plates
Steel shear transfer plates are provided at the intersection of the barrel penetration sleeve and liner, for both equipment and personnel accesses, as shown in Figure 29. The shear
plates, made of A516 Gr. 70 steel, are designed to resist a total shear force produced by an
internal pressure of 1.5 accident pressure acting outward from the barrel. The shear force is
transferred from the barrel penetration sleeve to the liner and concrete by means of the shear transfer plates. It is conservatively assumed that the welded tee joint, at the
intersection of the liner and barrel penetration, does not resist any of the shear force.
5.8 Radial
Reinforcement
Draped horizontal and vertical tendons produce radial forces which act toward the opening as shown in Figures 11 and 12. These radial forces result in producing tensile stresses, in
the concrete, in the vicinity of the curved tendons. Steel reinforcement is provided in the
form of hook and straight bars to resist radial forces, as can be seen in Drawing D-421-036.
Hoop reinforcing resist the radial forces pro- duced by curved vertical tendons while
meridional reinforcing resist radial forces produced by curved horizontal tendons.
The unit radial forces and total radial forces are given by the following equations:
Unit Radial Force p = T/r N
Total Radial Force p = Ti Li i=l ri
p = unit radial force kips/inch P = total radial force KIPS T = initial prestressing tendon force KIPS r = radius of curvature of tendon N = total number of curved tendons
5.9 Summary
of Design and Conclusions
In selecting the analytical method used for determining the stress-resultant and stress-couple distributions in the shell under all critical loading conditions an extensive
bibliographic search was conducted and available methods were evaluated. In our judgment
the analysis of the stresses around the openings for the Three Mile Island Reactor Building
has been based on the most satisfactory of available methods.
The design was guided by the basic proposition that the best reinforcement is in fact the least reinforcement that will satisfy the requirement for carrying the shell loads around the
opening and the normal shear applied along the opening edge into the shell out to a
distance from the opening until a membrane state of stress is reached. Although not directly
applicable the IITRI studies on steel containment stru ctures conclusively showed that a stiff reinforcement around openings substantially reduces the burst strength of circular plates (Reference 30). In our judgment this design as evidenced by the data included in Section 5
provides the required reinforcement strengt h and further conservatism in determining reinforcement requirements is not prudent in t hat the ultimate load capacity might be thereby reduced.
TMI-1 UFSAR APPENDIX 5C 5C-23 REV. 18, APRIL 2006 LIST OF REFERENCES
- 1. A. I. Lur'e: "Concentration of Stresses in the Vicinity of an Aperture in the Surface of a Circular Cylinder," Prikl. Mat. Mekh. 10 , 1946, pp. 397-407, (English translation by N.
Brunswick, N.Y. Univ., Inst. of Math. Science, 1960)
- 2. S. Timoshenko: "Strength of Materials , Part II," D. Van Nostrand Company, Inc., New York, N.Y., 1930, p. 457.
- 3. K. Sezawa and K. Kubo: "Stresses in a Plate with a Flanged Circular Hole," Report of the Aeronautical Research Institute , Tokyo Imperial University, Tokyo, Japan, No. 84, September, 1932.
- 4. G. Gurney: "Analysis of the Stresses in a Flate Plate with a Circular Hole Under Edge Forces," British Rep. Memo No. 1834 , February, 1938.
- 5. Leon Beskin: "Strengthening of Circular Holes in Plates Under Edge Loads," Journal of Applied Mechanics , Trans. ASME, Vol. 66, 1944, p. A-140.
- 6. G. N. Savin: "Stress Concentration Around Holes," Pergamon Press, 1961.
- 7. D. Withum: "Die Kreiszylinderschale mit Kreisformigem Ausschnitt unter Schubbean-Spruchung," Ingerieur Archiv, XXVI, 1958, pp. 435-446.
- 8. V. Kline, R. C. Dixon, N. F. Jordan, and A. C. Eringen: "Stresses in Pressurized Cylindrical Shell with Cut-out," General Technology Corp. Tech, Report 3-1 , 1961.
- 9. A. C. Eringen, A. K. Naghdi, and C. C. Thiel: "State of Stress in a Circular Cylindrical Shell with a Circular Hole," Welding Research Council Bulletin 102 , 1965.
- 10. A. K. Naghdi and A. C. Eringen: "Stress Distribution in a Circular Cylindrical Shell with a Circular Cut-out," Ingenieur Archiv.
- 11. J. G. Lekkerkerker: "Stress Concentration Around Circular Holes in Cylindrical Shells," Eleventh Int. Congress on Applied Mechanics , Munich, 1964.
- 12. FELAP - Finite Element Analysis Program for Complex Structures, Franklin Institute Research Laboratories , Philadelphia, Pa.
- 13. Z. Zudans: "New Formulation and Evaluation of Elastic Shell Theory
," Ph. D. Dissertation, University of Pennsylvania, May 1966.
- 14. Z. Zudans: "Analysis of Asymmetric Stiff ened Shell Type Structures by the Finite-Element Method," paper to be published in "Nuclear Engineering and Design
."
- 15. J. J. Connor and C. Brebbia: "Stiffness Matrix for Shallow Rectangular Shell Elements", ASCE Journal , Vol. 93, EM5, pp. 43-65, 1967.
TMI-1 UFSAR APPENDIX 5C 5C-24 REV. 18, APRIL 2006
- 16. C. Prato: "A Mixed Finite-Element Method for Shell Analysis
," Ph.D. Dissertation, M.I.T., June 1968.
- 17. F. Yao "Finite-Element Method for Shell Analysis," Ph.D. Dissertation, M.I.T., January 1968.
- 18. M. M. Leven: "Photoelastic Determination of Stresses in Reinforced Openings in Pressure Vessels," Welding Research Council Bulletin No. 113 , N.Y., 1966, pp. 25-52.
- 19. C. E. Taylor and N. C. Lind: "Photoelastic St udy of the Stresses Near Openings in Pressure Vessels," ibid., pp. 1-24.
- 20. S. K. Takahashi and R. Mark: "Analysis and Photoelastic Investigations of Stress Concentrations in Sphere-Cylinder Configurations
," U. S. Naval Civil Engineering Laboratory Report.
- 21. A. J. Durelli, D. J. del Rio, V.J. Parks, H. Feng: "Stresses in a Pressurized Cylinder with a Hole," Journal ASCE , Vol. 93, ST5, 1967, pp. 383-399.
- 22. R. Mark and J. D. Riera: Discussion of Ref. (19), Journal ASCE , Vol. 94, ST4, pp. 1075-76.
- 23. J. L. Mershon: "Preliminary Evaluation of PVCR Photoelastic Test Data on Reinforced Openings in Pressure Vessels," Welding Research Council Bulletin No. 113 , 1966.
- 24. D. S. Houghton and A Rothwell: "The Influence of Curvature on the State of Stress Ahead of Cracks in Cylindrical Shells," Non-Classical Shell Problems , Proc. of the Int. Symposium on Non-Classical Shell Problems, Warsaw, Poland, 1963, North Holland, 1964, pp. 685-700.
- 25. R. Mark: "Photomechanical Model Analysis of Concrete Structures," ACI Symposium on Models , 1968.
- 26. J. H. Barnwell and C. H. Edwards: "The Development and Testing of Three-Dimensional Photoelastic Models of Pretensioned, Prestressed Concrete Beams," paper presented at the
1968 SESA Spring Meeting , May 1968, Albany, N.Y.
- 27. Ch. H. Lerchenthal: "A Photoelastic Investigation of Stress Concentration Around Reinforced Cut-outs in Stresses Sheets," ibid.
- 28. R. D. Brunell "Computer Program Heating and Heating 2," Atomic International Division Report No. AI-64-MEMO-177, North American Aviation Inc., Detroit, Michigan, USA, 1964.
- 29. R. F. Mast: "Auxiliary Reinforcement in Concrete Connections," Journal ASCE , Vol. 94, No.
ST6, June 1968, pp. 1485-1504.
- 30. M. A. Salmon: "Studies of Reactor Containment Structures," IITRI Final Report for U.S.
AEC, Dept. of Commerce, Washington, D. C., 1966.
TMI-1 UFSAR APPENDIX 5C 5C-25 REV. 18, APRIL 2006
TABLES
TMI-1 UFSAR APPENDIX 5C 5C-1 REV. 18, APRIL 2006 FUNDAMENTAL LOADS TABLE 5C-1 Load No. DL (1) H&VP (2) OT (3) AT (4) IP (5) E 1 (6) E 2 (7) (8) 1 1.000 1.260 0.0 0.0 0.0 0.0 0.0 0.0 2 1.000 1.110 0.270 0.0 0.0 0.0 0.0 0.0 3 1.000 1.110 1.000 0.0 0.0 0.0 0.0 0.0 4 1.000 1.110 0.270 0.0 0.0 0.625 0.0 0.0 5 1.000 1.110 1.000 0.0 0.0 0.625 0.0 0.0 6 1.000 1.110 0.270 0.0 0.0 -0.625 0.0 0.0 7 1.000 1.110 1.000 0.0 0.0 -0.625 0.0 0.0 8 1.000 1.110 0.270 0.0 0.0 0.0 0.625 0.0 9 1.000 1.110 1.000 0.0 0.0 0.0 0.625 0.0 10 1.000 1.110 0.270 0.0 0.0 0.0 -0.625 0.0 11 1.000 1.110 1.000 0.0 0.0 0.0 -0.625 0.0 12 1.000 1.000 0.270 0.0 0.0 0.0 0.0 0.0 13 1.000 1.000 1.000 0.0 0.0 0.0 0.0 0.0 14 1.000 1.000 0.270 0.0 0.0 0.625 0.0 0.0 15 1.000 1.000 1.000 0.0 0.0 0.625 0.0 0.0 16 1.000 1.000 0.270 0.0 0.0 -0.625 0.0 0.0 17 1.000 1.000 1.000 0.0 0.0 -0.625 0.0 0.0 18 1.000 1.000 0.270 0.0 0.0 0.0 0.625 0.0 19 1.000 1.000 1.000 0.0 0.0 0.0 0.625 0.0 20 1.000 1.000 0.270 0.0 0.0 0.0 -0.625 0.0 Normal Operation 21 1.000 1.000 1.000 0.0 0.0 0.0 -0.625 0.0 22 1.000 1.110 0.270 0.0 1.150 0.0 0.0 0.0 23 1.000 1.110 1.000 0.0 1.150 0.0 0.0 0.0 24 1.000 1.000 0.270 0.0 1.150 0.0 0.0 0.0 Test Condition 25 1.000 1.000 1.000 0.0 1.150 0.0 0.0 0.0 26 1.000 1.110 0.270 1.000 1.000 0.0 0.0 0.0 27 1.000 1.110 1.000 1.000 1.000 0.0 0.0 0.0 28 1.000 1.110 0.270 1.000 1.000 1.000 0.0 0.0 29 1.000 1.110 1.000 1.000 1.000 1.000 0.0 0.0 30 1.000 1.110 0.270 1.000 1.000 -1.000 0.0 0.0 31 1.000 1.110 1.000 1.000 1.000 -1.000 0.0 0.0 32 1.000 1.110 0.270 1.000 1.000 0.0 1.000 0.0 33 1.000 1.110 1.000 1.000 1.000 0.0 1.000 0.0 34 1.000 1.110 0.270 1.000 1.000 0.0 -1.000 0.0 35 1.000 1.110 1.000 1.000 1.000 0.0 -1.000 0.0 36 1.000 1.000 0.270 1.000 1.000 0.0 0.0 0.0 37 1.000 1.000 1.000 1.000 1.000 0.0 0.0 0.0 38 1.000 1.000 0.270 1.000 1.000 1.000 0.0 0.0 39 1.000 1.000 1.000 1.000 1.000 1.000 0.0 0.0 40 1.000 1.000 0.270 1.000 1.000 -1.000 0.0 0.0 Loading Combinations Accident Pressure Loadings 41 1.000 1.000 1.000 1.000 1.000 -1.000 0.0 0.0 TMI-1 UFSAR APPENDIX 5C 5C-2 REV. 18, APRIL 2006 FUNDAMENTAL LOADS (continued)
TABLE 5C-1 (continued)
Load No. DL (1) H&VP (2) OT (3) AT (4) IP (5) E 1 (6) E 2 (7) (8) 42 1.000 1.000 0.270 1.000 1.000 0.0 1.000 0.0 43 1.000 1.000 1.000 1.000 1.000 0.0 1.000 0.0 44 1.000 1.000 0.270 1.000 1.000 0.0 -1.000 0.0 45 1.000 1.000 1.000 1.000 1.000 0.0 -1.000 0.0 46 1.000 1.110 0.270 1.000 1.250 0.0 0.0 0.0 47 1.000 1.110 1.000 1.000 1.250 0.0 0.0 0.0 48 1.000 1.110 0.270 1.000 1.250 0.625 0.0 0.0 49 1.000 1.110 1.000 1.000 1.250 0.625 0.0 0.0 50 1.000 1.110 0.270 1.000 1.250 -0.625 0.0 0.0 51 1.000 1.110 1.000 1.000 1.250 -0.625 0.0 0.0 52 1.000 1.110 0.270 1.000 1.250 0.0 0.625 0.0 53 1.000 1.110 1.000 1.000 1.250 0.0 0.625 0.0 54 1.000 1.110 0.270 1.000 1.250 0.0 -0.625 0.0 55 1.000 1.110 1.000 1.000 1.250 0.0 -0.625 0.0 56 1.000 1.000 0.270 1.000 1.250 0.0 0.0 0.0 57 1.000 1.000 1.000 1.000 1.250 0.0 0.0 0.0 58 1.000 1.000 0.270 1.000 1.250 0.625 0.0 0.0 59 1.000 1.000 1.000 1.000 1.250 0.625 0.0 0.0 60 1.000 1.000 0.270 1.000 1.250 -0.625 0.0 0.0 61 1.000 1.000 1.000 1.000 1.250 -0.625 0.0 0.0 62 1.000 1.000 0.270 1.000 1.250 0.0 0.625 0.0 63 1.000 1.000 1.000 1.000 1.250 0.0 0.625 0.0 64 1.000 1.000 0.270 1.000 1.250 0.0 -0.625 0.0 65 1.000 1.000 1.000 1.000 1.250 0.0 -0.625 0.0 66 1.000 1.000 0.270 1.000 1.150 0.0 0.0 0.0 67 1.000 1.260 1.000 1.000 1.150 0.0 0.0 0.0 68 1.000 1.000 0.270 1.000 1.150 0.0 0.0 0.0 Accident Pressure Loadings (continued) 69 1.000 1.000 1.000 1.000 1.150 0.0 0.0 0.0 70 1.000 1.110 0.270 0.0 0.0 1.000 0.0 0.0 71 1.000 1.110 1.000 0.0 0.0 1.000 0.0 0.0 72 1.000 1.110 0.270 0.0 0.0 -1.000 0.0 0.0 73 1.000 1.110 1.000 0.0 0.0 -1.000 0.0 0.0 74 1.000 1.110 0.270 0.0 0.0 0.0 1.000 0.0 75 1.000 1.110 1.000 0.0 0.0 0.0 1.000 0.0 76 1.000 1.110 0.270 0.0 0.0 0.0 -1.000 0.0 Loading Combinations (continued)
Normal Operations 77 1.000 1.110 1.000 0.0 0.0 0.0 -1.000 0.0 DL = Dead Load AT = Accident Temperature Note: Coefficients are based on: H & VP = Horizontal and Vertical Prestress IP = Internal Pressure Accident Pressure, Winter Temperature, OT = Operating Temperature E1 = Earthquake 1 Final Prestress, Maximum Hypothetical Earthquake E2 = Earthquake 2 TMI-1 UFSAR APPENDIX 5C 5C-1 REV. 18, APRIL 2006 TABLE 5C-2 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 1 DEAD LOAD MEMBRANE TWISTING AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q 11 -5.55 14.78 -0.15 3.58 0.12 -0.36 0.42 0.00
22 -5.05 -17.67 -1.05 -0.60 0.21 -0.25 0.46 -0.01
33 -4.61 -59.53 -0.94 -10.57 0.43 5.09 0.27 0.26
44 -2.06 -8.70 1.36 25.65 0.79 15.23 -1.26 0.10
55 -0.50 8.10 4.01 77.44 0.71 10.54 -0.66 -0.54
66 0.30 13.15 7.17 139.22 0.45 3.47 -0.48 -1.01
77 0.44 4.67 12.26 241.50 -0.63 -16.09 0.65 -0.88
25 -8.70 -22.61 0.78 3.48 -0.32 -9. 85 0.23 0.08
49 -12.24 -210.41 2.13 64.37 -0.28 -16.70 0.82 -0.53
73 -14.62 -259.86 1.26 48.47 0.30 -10.33 0.86 -0.25
74 -11.32 -198.03 0.58 10.96 2.43 31.48 0.42 0.86
94 -8.11 11.44 0.79 -8.53 -0.12 8.44 -2.32 -0.12
97 -11.96 -142.28 -1.19 5.85 -0.30 5.98 0.03 0.43
99 -16.70 -279.92 -3.20 -45.84 -0.38 -10.72 0.11 1.25
100 -21.34 -366.85 -3.39 -56.70 0.28 0.89 1.12 1.51
101 -29.35 -504.98 -1.93 -11.53 2.56 48.08 2.72 -2.36
NOTE: N, N, N, Q, and Q in Kips/in.
M and M in Kips-in/in. Values computed by finite-element analysis (Sheet 1 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-2 REV. 18, APRIL 2006 TABLE 5C-3 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 2 HOOP AND VERTICAL PRESTRESS MEMBRANE TWISTING AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q
11 -31.79 -142.30 -72.32 55.63 0.14 -1.20 -0.44 0.12
22 -29.51 -118.97 -67.36 61.36 0.73 -2.52 -0.35 0.05
33 -24.90 -167.75 -91.91 -789.23 2.69 12.76 -0.50 32.16
44 -8.42 2.42 -113.30 -1906.16 6.02 64.67 -3.67 2.62
55 -1.13 -13.56 -112.86 - 1752.00 9.22 107.44 -10.39 2.84
66 -9.01 -80.17 -118.33 -1631.09 10.90 122.42 -18.66 0.59
77 -10.19 -17.42 -125.59 -1570.90 14.17 124.75 -27.16 3.44
25 -39.83 -117.50 -74.70 -241.10 1.72 -150.89 2.59 4.00
49 -45.42 -379.25 -69.20 -751.08 16.13 33.18 0.22 3.20
73 -52.07 -458.64 -53.36 -659.38 29 .47 224.47 0.31 2.26
74 -53.81 -488.44 -60.14 -809.52 42.35 392.40 10.25 6.37
94 -39.10 47.94 -55.55 9.88 0.52 -4.06 0.73 3.52
97 -60.31 -475.61 40.43 -255.04 2.58 22.90 -0.42 -4.61
99 -68.06 -543.16 -17.65 -86.70 7.30 84.43 2.09 -6.53
100 -72.22 -416.40 -6.67 -30.10 9.89 93.83 -1.10 10.62
101 -76.92 -219.28 -7.43 26.58 10.27 46.46 -4.02 14.42
NOTE: N, N, N, Q, and Q in Kips/in.
M and M in Kips-in/in.
Values computed by finite-element analysis (Sheet 2 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-3 REV. 18, APRIL 2006 TABLE 5C-4 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 3 OPERATING TEMPERATURE (WINTER)
MEMBRANE TWISTING AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q
11 -3.10 514.41 10.23 433.38 -0.17 0.10 0.20 -0.01
22 -3.96 529.61 7.96 440.66 -0.12 -0.61 -0.69 -0.02
33 0.24 694.27 0.39 984.76 0.11 -7.74 -5.00 -33.98
44 -4.29 490.90 -14.95 1582.80 0.27 -38.15 -1.04 0.08
55 -4.07 323.86 -18.51 1652.76 0.08 -97.75 1.43 -3.91
66 -3.40 268.22 -22.05 1771.96 0.25 -132.39 3.86 1.59
77 -1.16 145.44 -25.85 1932.96 2.60 -156.72 3.68 12.47
25 0.16 514.79 2.91 571.08 -3.92 101.63 -1.85 -2.50
49 -5.83 1144.06 -11.13 759.89 1.09 -38.77 -1.25 0.41
73 -6.41 1346.10 -6.66 730.25 4.13 -294.50 5.43 0.54
74 -5.76 1196.31 -7.02 955.61 4.50 -606.81 -3.38 -3.62
94 -17.97 489.90 -6.49 479.64 -4.98 102.04 -24.95 -0.16
97 -13.53 924.93 -4.45 642.63 0.50 -2.91 0.46 9.88
99 -8.48 1683.20 -3.26 478.10 0.57 -51.10 0.58 -1.90
100 -9.16 1866.96 -2.10 341.98 0.68 -106.07 0.01 -3.47
101 -12.72 2053.34 -0.91 35.57 1.61 -187.64 -0.52 7.54
NOTE: N, N, N, Q, and Q in Kips/in. M and M in Kips-in/in. Values computed by finite-element analysis (Sheet 3 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-4 REV. 18, APRIL 2006 TABLE 5C-5 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 4 ACCIDENT TEMPERATURE (WITH BARRELL EFFECT)
MEMBRANE TWISTING AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q 11 -2.86 217.29 0.09 186.63 -0.17 0.73 0.74 0.00
22 -3.74 170.86 0.68 176.19 -0.24 0.58 0.70 0.00
33 -4.34 140.97 4.97 333.57 -0.62 -8.70 0.40 -11.43
44 -9.82 -78.19 13.42 666.08 -1.80 -44.84 2.41 -0.05
55 -14.81 -278.23 17.75 787.17 -3.77 -108.26 3.15 -0.94
66 -20.71 -431.10 15.33 785.34 -6.40 -154.00 1.21 1.63
77 -27.65 -638.49 17.94 927.02 -10.08 -217.19 -2.52 18.04
25 -1.56 179.23 -1.02 205.89 -0.29 20.06 -0.44 -0.51
49 2.31 357.30 0.12 260.03 -4.03 -174.35 -1.44 -0.38
73 7.88 563.30 -0.85 185.09 -10.00 -353.01 -2.05 -0.07
74 0.99 356.46 0.62 274.74 -15.13 -534.57 0.72 0.32
94 -1.48 182.56 -1.67 168.19 -0.03 -0.75 0.08 -0.45
97 5.09 435.56 -4.30 164.76 -0.68 -17.07 -0.28 3.49
99 13.85 856.12 -12.86 129.17 -2.98 -87.04 -0.45 3.00
100 15.84 956.25 -20.01 -343.88 -6.20 -164.51 -0.64 5.65
101 7.86 892.45 -26.94 -613.86 -9.40 -255.04 16.46 -2.69
NOTE: N, N, N, Q, and Q in Kips/in.
M and M in Kips-in/in. Values computed by finite-element analysis (Sheet 4 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-5 REV. 18, APRIL 2006 TABLE 5C-6 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 5 55 psig INTERNAL PRESSURE
MEMBRANE TWISTING AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q 11 23.12 88.28 45.64 -37.45 -0.12 1.52 0.81 -0.04
22 22.51 35.40 43.89 -56.46 -0.13 2.98 1.00 -0.02
33 22.43 29.84 58.58 423.42 -0.39 6.13 -0.18 -18.10
44 18.98 -5.22 74.08 1070.76 -1.22 4.56 0.85 -0.06
55 14.66 -55.50 82.16 1157.09 -3.64 -54.47 6.79 -3.58
66 8.84 -90.60 93.92 1304.67 -6.74 -96.68 8.83 -3.47
77 -1.05 -194.66 111.53 1589.83 -12.51 -160.33 13.96 2.52
25 20.98 52.28 47.42 169.43 -0.63 78.06 -1.62 -2.49
49 23.20 115.32 47.20 639.42 -10.54 -98.54 -1.12 -2.14
73 25.75 267.38 36.81 532.50 -20.32 -324.05 -3.19 0.21
74 25.24 213.57 39.74 607.13 -28.21 -427.25 -12.22 -0.96
94 20.55 -38.89 39.23 -110.38 -0.20 -7.25 1.36 -3.85
97 34.65 531.60 29.36 346.81 -1.70 -21.80 -0.58 5.64
99 36.80 701.25 13.72 103.07 -4.49 -81.21 -1.10 5.31
100 34.78 625.07 3.72 -34.14 -6.12 -109.36 -0.53 7.93
101 29.14 475.54 -5.16 -172.79 -6.51 -115.10 6.09 6.90
NOTE: N, N, N, Q, and Q in Kips/in.
M and M in Kips-in/in.
Values computed by finite-element analysis (Sheet 5 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-6 REV. 18, APRIL 2006 TABLE 5C-7 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 6 EARTHQUAKE #1
MEMBRANE TWISTING AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q 11 -4.07 10.83 -0.11 2.62 0.09 -0.26 0.31 0.00
22 -3.70 -12.95 -0.77 -0.44 0.15 -0.18 0.34 -0.01
33 -3.38 -43.64 -0.69 -7.75 0.32 3.73 0.20 0.19
44 -1.51 -6.38 1.00 18.80 0.58 11.16 -0.92 0.07
55 -0.37 5.94 2.93 56.76 0.52 7.73 -0.48 -0.40
66 0.22 9.64 5.26 102.05 0.33 2.54 -0.35 -0.74
77 0.32 3.42 8.99 177.02 -0.46 -11.79 0.48 -0.65
25 -6.38 -16.57 0.57 2.55 -0.23 -7.22 0.17 0.06
49 -8.97 -154.23 1.56 47.18 -0.21 -12.24 0.60 -0.39
73 -10.72 -190.48 0.92 35.53 0.22 -7.57 0.63 -0.18
74 -8.30 -145.16 0.43 8.03 1.78 23.07 0.31 0.63
94 -5.94 8.39 0.58 -6.25 -0.09 6.19 -1.70 -0.09
97 -8.77 -104.29 -0.87 4.29 -0.21 -4.38 0.02 0.32
99 -12.24 -205.18 -2.35 -33.60 -0.29 -7.86 0.08 0.92
100 -15.64 -268.90 -2.48 -41.56 0.21 0.65 0.82 1.11
101 -21.51 -370.15 -1.41 -8.45 1.88 35.24 1.99 -1.73
NOTE: N, N, N, Q, and Q in Kips/in.
M and M in Kips-in/in.
Values computed by finite-element analysis (Sheet 6 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-7 REV. 18, APRIL 2006 TABLE 5C-8 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 7 EARTHQUAKE #2 MEMBRANE TWISTING
AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q 11 -0.56 1.48 -0.02 0.36 3.59 -0.04 0.04 0.00
22 -0.51 -1.77 0.69 -0.06 2.72 -0.03 0.05 0.00
33 -0.46 -5.95 1.41 -1.06 2.04 0.51 0.03 0.03
44 -0.21 -0.87 2.14 2.57 1.58 1.52 -0.13 0.01
55 -0.05 0.81 2.90 7.74 1.07 1.05 -0.07 -0.05
66 0.03 1.32 3.72 13.92 0.55 0.35 -0.05 -0.10
77 0.04 0.47 4.81 24.15 -0.06 -1.61 0.07 -0.09
25 -0.87 -2.26 0.08 0.35 3.55 -0.99 0.02 0.01
49 -1.22 -21.04 0.21 6.44 3.55 -1.67 0.08 -0.05
73 5.54 -25.99 7.13 4.85 7.03 -1.03 0.09 -0.03
74 7.77 -19.80 8.96 1.10 9.14 3.15 0.04 0.09
94 -0.81 1.14 0.08 -0.85 3.57 0.84 -0.23 -0.01
97 -0.20 -14.23 -0.12 0.59 2.47 -0.60 0.00 0.04
99 0.13 -27.99 -0.32 -4.58 1.76 -1.07 0.01 0.13
100 0.37 -36.69 -0.34 -5.67 1.03 0.09 0.11 0.15
101 0.64 -50.50 -0.19 -1.15 0.26 4.81 0.27 -0.24
NOTE: N, N, N, Q, and Q in Kips/in.
M and M in Kips-in/in.
Values computed by finite-element analysis (Sheet 7 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-8 REV. 18, APRIL 2006
FIGURES
TMI-1 UFSAR APPENDIX 5C 5C-9 REV. 18, APRIL 2006
DRAWINGS
TMI-1 UFSAR APPENDIX 5C 5C-10 REV. 18, APRIL 2006
APPENDIX A Effect of Concrete Creep and the Sustained Operating Stresses on Stress Distribution Around Openings in a Rapidly Pressurized Reinforced Concrete Vessel.
TMI-1 UFSAR APPENDIX 5C 5C-11 REV. 18, APRIL 2006 APPENDIX A EFFECT OF CONCRETE CREEP AND THE SUSTAINED OPERATING STRESSES ON THE STRESS DISTRIBUTION AROUND OPENINGS IN A RAPIDLY PRESSURIZED REINFORCED CONCRETE VESSEL
Consider the simple structure shown below:
Figure A-1
Column 1 is a reinforced concrete column with net concrete area A 1 C and longitudinal steel areas A st 1. A 2 C and A 2 st denote the net concrete area and the logitudinal steel area, respectively, of reinforced concrete columns 2. The system is loaded at time t = 0 with a vertical load P. Let us
determine the initial load distribution:
P = T 1 + 2T 2 sin 600 = T 1 + 3 T 2 (1)
delta L 2 = delta L 1 sin 600; delta L 2 = 3 delta L 1 (2) 2
in which L 1= T 1 (3) L A 1 C E C + A st E st 1 3 delta L 2 = T 2 (4) 2 L A
C 2 E C + A st 2 E st E C and E st denote the "effective" modulae of elasticity of concrete and steel, T 1 and T 2 the loads carried by columns 1 and 2, respectively. From equations (1) to (4) we obtain:
T 1 + 3 T 2 = P 3 delta L T 1 - 2 L T 2 = 0 2 1 1 3 A C E C + A st E st A C 2 E C + A 2 st E st
or T 1 J 1.5I + J
= P
{ { 3 I
} T 2 2 1.5I + J (5)
TMI-1 UFSAR APPENDIX 5C 5C-12 REV. 18, APRIL 2006 In which I = L
A C 1 E C + A st 1 E st J = 2 L
3 A
C 2 E C + A st 2 E st (6)
Equation (5) gives the loads acting on Columns 1 and 2 in terms of the "effective" modulae E C and E st In general, T 1 and T 2 will change with time (Load redistribution) due to concrete creep. Under the assumption that concrete behaves like a Kelvin-type material, the load distribution may be
calculated exactly for t ¥ by resorting to the creep-limit modulus E cu. (Time-dependent behavior of steel is neglected).
Note that even when there is no load redistribution (for example, when I = B) there will be some stress redistribution. In other words, T 1 an T 2 may remain constant, but the percentage of both carried by the steel reinforcement will increase as concrete creeps. As a result, steel stresses will increase and concrete stresses will decrease to final values which may be easily computed.
Let us now assume that at time t 1 a load P 1 issuperimposed on the existing load P 8 giving a total load (P 8 + P 1). (See Figure A-1). let T 1 and T 2 be the column loads immediately before the load P 1 is applied. To compute the actual column loads T 1 and T 2 after P 1 is applied, we would be tempted to determine the column loads T 1* and T 2* corresponding to P 1 acting alone on the structure (On the basis of the initial modulus of elasticity) and then add them to T 1 1 and T 2 1:
T 1 2 = T 1 1 + T 1* (7)
T 2 2 = T 2 1 + T 2* (8)
The approach is valid if there is no concrete cracking. If either column 1 or 2 (or both) cracks due to the resulting tensile stresses, the results obtained by restoring to equation (7) will be incorrect. The
situation will be best illustrated by an example. Let:
E C o = initial modulus of elasticity of concrete = 4000 ksi
E u C = "final" effective modulus of elasticity of concrete = 2000 ksi
E st = 7.5E C o = 30000 ksi
A C 1 = 100 in.
2 TMI-1 UFSAR APPENDIX 5C 5C-13 REV. 18, APRIL 2006 Ac 2 = 100 in.
2 A st 1 = 2 in.2 A st 2 = 4 in.2 L = 1000 in.
A load P 8 = -200 kips is applied at T = 8 and kept constant.
At time t = t 1 a second load P 1 = +300 kips is applied.
Initial Load Distribution under P = P 8
i = 1000 1 = 8.69 100 + 2 x 7.5 E C 8 E C 8 (9) J i = 2 1000 1 = 8.875 3 100 + 4 x 7.5 E C 8 E C 8 (10) T 1 8 = 8.875 P 8 = 0.404P 8 1.5 x 8.69 + 8.875 T 2 8 = 8.69 x 0.866 P 8 = 0.343P 8 (11) 1.5 x 8.69 + 8.875
Load Distribution under P = P 8 Immediately Before Application
of P 1 (t 1 ¥)
f = 1000 1 = 15.4 0.5 x 100 + 2 x 7.5 E C 8 E C 8 (12)
J f = 1.155 x 1000 1 = 14.45 0.5 x100 + 4 x 7.5 E C 8 E C 8 1 T 11 = 14.45 P 8 = 0.385P 1.5 x 15.4 + 14.45
1 T 2 = 15.4 x 0.866 P 8 = 0.355P 8 1.5 x 15.4 + 14.45 (13)
Concrete Stresses Due to P = P 8
8 f
C1 = Ii T 1E 8 e = -704 psi Initial Stresses L
{
8 8 f C2 = Ji T 2 E C = -608 psi L (14)
TMI-1 UFSAR APPENDIX 5C 5C-14 REV. 18, APRIL 2006 1-u f C1 = If T l E C = -598 psi Final Stresses L
(t 1 ¥ ) { 1-u f C2 = Jf T 2E C = -514 psi L (15)
Consequently, if t 1 is large, concrete stresses immediately after P 1 is applied would be:
f C 1+1 = -598 + 1055 = +457 psi
f C 1+2 =-514 + 912 = +398 psi (16)
If the tensile strength of concrete is f t = 420 psi, then column 1 should be expected to crack, i.e., the entire load T 1 would be carried by the steel reinforcement. Under such conditions, a load redistribution would occur, resulting in a large increase in T 2 , with subsequent cracking of columns 2 as well. The final load distribution will therefore depend on A st 1 and A st 2 only. That is to say, if both columns crack it is irrelevant whether t 1 is small or large. Moreover, A C 1 and A C 2 will no longer play any role in the problem. In fact:
I 1+ = 1000
= 0.01667 2 x 30000
J 1+ = 1.155 x 1000
= 0.00962 (17) 4 x 30000
T 1 1+ = 0.00962 P
= 0.278P 1.5 x 0.01667 + 0.00962
T 2 1+ = 0.01667 x 0.866
= 0.416P 1.5 x 0.01667 + 0.00962 (18)
It has been shown that if our sample structure is fully cracked, then creep has no influence
whatsoever on the final load distribution. It may be hypothesized, however, that the concrete
tensile strength in Column 2 is higher, say f t = 1500 psi. The question is than asked, what is now the load distribution. The problem may be solved by computing the total load "C" carried by concrete in Column 1 at time shortly before t = t 1 , and assuming that, as Column 1 cracks, load will be transferred simultaneously to joint and to the steel of column 1.
Initial Load Distribution with Concrete in Column 1 Fully Cracked
I c = 1000 1 = 66.6 2 x 7.5 E C 8 E C 8 J 8 = 1.155 x 1000 8.875 100 + 4 x 7.5 E C 8 (19)
T 1 = 8.875 P = 0.082 P 1.5 x 66.6 + 8.875
TMI-1 UFSAR APPENDIX 5C 5C-15 REV. 18, APRIL 2006 T 2 = 0.866 x 66.6 P = 0.531 P 1.5 x 66.6 + 8.875 (20)
T 1 = 0.385 P 8 + 0.082 (P 1 + C) - C
T 2 = 0.355 P 8 + 0.531 (P 1 + C) (21)
with C = -.598 x 100 = -59.8 kips we get:
T 1 = +2.6 kips t 1 ¥ T 2 = +56.9 kips } (22)
If t 1 is sufficiently small, it may be assumed that P 8 and P 1 are applied simultaneously at t = 0, in which case:
T 1 = 0.082 x 100 = +8.2 kips t 1 = 0 T 2 = 0.531 x 100 = +53.1 kips } (23)
The difference between (22) and (23) is not large. Note that superimposing the load distribution
after creep due to P 8 (equation 13) with the load distribution corresponding to the cracked structure under P 1 leads to:
T 1 =-0.385 x 200 + 0.082 x 300 = - 42.4 kips T 2 =-0.355 x 200 + 0.531 x 300 = 88.3 kips (24)
which are entirely unrealistic figures. Note also t hat an "exact" stress analysis for the case when t 1 is large (equations (19) and (20)) would not be feasible for moderately complex structures.
TMI-1 UFSAR APPENDIX 5C 5C-16 REV. 18, APRIL 2006
APPENDIX B Earthquake Analysis
TMI-1 UFSAR APPENDIX 5C 5C-17 REV. 18, APRIL 2006 APPENDIX B
EARTHQUAKE ANALYSIS
The computation of seismic stresses was carried out on the basis of the fundamental mode of the
containment structure associated with maximum response. The peak of the response curve (0.31g)
for 2 percent critical damping and 0.12g peak ground acceleration was used to determine:
(1) The stress-resultant Nf at the center of the opening (in the shell without the opening) for the horizontal component of earthquake motion oriented in the direction normal to
the openings.
(2) The in-plane shear stress-resultant NfP at the center of the opening (in the shell without the opening), for the horizontal component of earthquake motion oriented at
900 with the direction normal to the opening.
(3) The stress-resultant Nf at the center of the opening (in the shell without the opening) for the vertical component of motion associated with 0.12g peak ground acceleration.
The influence of the opening on the above seismic loads was evaluated as follows:
- a. The stress-resultant and stress-couple distributions and, therefore, the stress-concentration factors corresponding to (1) and (3) were conservatively
computed on the basis of the finite-element results for dead load.
- b. The stress concentration factors corresponding to (2) were determined on the basis of Lekerkkerker's solution (Reference 11) (See Figure 3) for a shell with a hole
subjected to torsion, i.e., to a pure membrane shear NfP at the location of the
opening, (in the shell without the opening). Note that the stress concentration factors
are slightly larger than those corresponding to the plate solution. In computing
stresses at elements away from the edge of the opening, however, the stress concentration factor was assumed to decrease as in the plane solution.
In determining the values shown in Table (6) and (7) the absolute value of the contribution of the
vertical component of motion (3) was added to the absolute value of the contributions of the two
horizontal components [(1) and (2)].
TMI-1 UFSAR CHAPTER 05 5C-i REV. 18, APRIL 2006
APPENDIX 5C
GILBERT ASSOCIATES, INC., REPORT
TO METROPOLITAN EDISON COMPANY
ON DESIGN OF LARGE OPENING REINFORCEMENTS FOR REACTOR BUILDING
FOR THREE MILE ISLAND NUCLEAR STATION UNIT 1
This Report Contains:
44 pages of text 8 tables 29 figures 2 drawings Appendix A (8 pages)
Appendix B (3 pages)
TMI-1 UFSAR CHAPTER 05 5C-ii REV. 18, APRIL 2006
DESIGN OF LARGE OPENING REINFORCEMENTS FOR REACTOR BUILDING METROPOLITAN EDISON COMPANY THREE MILE ISLAND NUCLEAR POWER STATION - UNIT NO. 1
BY GILBERT ASSOCIATES, INC.
TMI-1 UFSAR CHAPTER 05 5C-iii REV. 18, APRIL 2006 TABLE OF CONTENTS
ITEM TITLE
SUMMARY
1 DESIGN BASES
1.1 General
1.2 Design
Loads 1.3 Load Combinations
1.4 Material
Stress/Strain Criteria 1.5 Test Condition
1.6 Operating
Condition
2 GENERAL DESCRIPTION OF OPENING REINFORCEMENT
2.1 Introduction
2.2 Ring Reinforcement Splices
2.3 Normal
Shear at Edge of Opening
2.4 Prestressing
- 3. STRESS DISTRIBUTION AROUND A CIRCULAR HOLE IN A CIRCULAR CYLINDRICAL SHELL 3.1 Introduction 3.2 Finite-Element Method
3.3 Application
of Three-Dimensional Photoelasticity
4 ANALYSIS OF THE STRESS AROUND LARGE OPENINGS IN THE THREE MILE ISLAND REACTOR BUILDING 4.1 Verification of Finite-Element Method of Analysis
4.2 Stress
Analysis Under Operating and Accident Loads
4.2.1 Representation
of the Shell Around the Opening
4.2.2 Basic
Loading Conditions
4.2.3 Effect
of Creep and Shrinkage
5 VERIFICATION OF DESIGN CRITERIA
5.1 Basis
for Verification of Shell Loading Capacity Due to Primary Loads (Principal Stress-resultants and Principal Stress-couples) 5.2 Interaction Diagram
5.3 Reinforcing
Steel
5.4 Penetration
Barrel
5.5 Concrete
Shear
5.6 Shear
- Diagonal Tension
5.7 Shear
- Transfer Plates
5.8 Radial
Reinforcement
5.9 Summary
of Design and Conclusion
LIST OF REFERENCES
TABLES
FIGURES
DRAWINGS TMI-1 UFSAR CHAPTER 05 5C-iv REV. 18, APRIL 2006 LIST OF TABLES
NUMBER TITLE 5C-1 Fundamental Loads
5C-2 Stress Around Equipment Hatch Loading Condition No. 1 Dead Load
5C-3 Stress Around Equipment Hatch Loading Condition No. 2 Hoop & Verticle Prestress
5C-4 Stress Around Equipment Hatch Loading Condition No. 3 Operating Temperature (Winter) 5C-5 Stress Around Equipment Hatch Loading Condition No. 4 Accident Temperature (with Barrel Effect)
5C-6 Stress Around Equipment Hatch Loading Condition No. 5 55 Psig Internal Pressure 5C-7 Stress Around Equipment Hatch Loading Condition No. 6 Earthquake # 1
5C-8 Stress Around Equipment Hatch Loading Condition No. 7 Earthquake #2 TMI-1 UFSAR CHAPTER 05 5C-v REV. 18, APRIL 2006 LIST OF FIGURES
NUMBER TITLE 5C-1 Total Stress Concentration Factors For Given Loading Cases
5C-2 Stress Concentration Factor Variations For Different Bpo At Various Angles -
Capped Cylinder
5C-3 Stress Distribution Around Openings in Cylindrical Shells
5C-4 Grid For Finite Element Analysis of Stresses Around Equipment Hatch In RCB
5C-5 Layer Thickness And Designation
5C-6 Member Stresses Around Opening Edge (Vessel Subject To Internal Pressure)
5C-7 Surface Stresses Around Opening Edge (Vessel Subject To Internal Pressure)
5C-8 Hoop Stresses Along Longitudinal Axis (Vessel Subject To Internal Pressure)
5C-9 Axial Stresses Along Transverse Axis (Vessel Subject To Internal Pressure)
5C-10 Hoop Stress-Resultant N o Along Symmetry Axes (Test Problem)
5C-11 Effect of Tendon Curvature
5C-12 Nodal Forces Due to Curvature of Tendons In Neighborhood of Operating
5C-13 Normal Winter Operating Temperat ure Gradient-84" Thick Wall Near Equipment Opening 5C-14 Normal Winter Operating Temperat ure Gradient-42" Thick Wall Near Equipment Opening 5C-15 Normal Winter Operating Temperatur e Gradient-Transition Wall Near Equipment Opening 5C-16 Winter Accident Temperature Gr adient-84" Thick Wall Near Equipment Opening
5C-17 Winter Accident Temperature Gr adient-42" Thick Wall Near Equipment Opening
5C-18 Winter Accident Temperature Gr adient-Transition Wall Near Equipment Opening
5C-19 Interaction Diagram For Axial Compression/Tension And Bending
5C-20 Equipment Access Interaction Diagram -Elements 77 & 101
5C-21 Equipment Access Interaction Diagram - Elements 66 & 100
5C-22 Equipment Access Interaction Diagram - Elements 55 & 99 TMI-1 UFSAR CHAPTER 05 5C-vi REV. 18, APRIL 2006 5C-23 Equipment Access Interaction Diagram - Element 44
5C-24 Equipment Access Interaction Diagram - Element 73
5C-25 Equipment Access Interaction Diagram - Element 74
5C-26 Equipment Access Interaction Diagram - Elements 66 & 77
5C-27 Equipment Access Interaction Diagram - Elements 44 & 55
5C-28 Equipment Access Interaction Diagram - Elements 99, 100, & 101
5C-29 Sheer Transfer Plates TMI-1 UFSAR CHAPTER 05 5C-vii REV. 18, APRIL 2006 APPENDIX A
Effect of Concrete Creep and the Sustained Operating Stresses on Stress Distribution Around Openings in a Rapidly Pressurized Reinforced Concrete Vessel
APPENDIX B
Earthquake Analysis
TMI-1 UFSAR APPENDIX 5C 5C-1 REV. 18, APRIL 2006
SUMMARY
DESIGN BASES
The Reactor Building large openings, consisting of a 22 feet 4 inches diameter opening (equipment
access hatch) and a 10 feet 0 inches diameter opening (personnel lock), are designed generally in
accordance with the criteria as detailed in The Three Mile Island Nuclear Station Unit 1 Safety
Analysis Report. The steel reinforcements used in the vicinity of the openings have a minimum yield stress of 40,000 psi.
GENERAL DESCRIPTION
The vertical and horizontal prestressing tendons are draped around the holes and are continuous (i.e., no tendons are terminated at the hole). Circular rebar rings are located around the hole as
principal reinforcement. Horizontal and verical rebars are provided to the extent required by calculated radial (to the hole) stresses. Normal shear forces, due to pressure on the hatch, at the
interface between penetration barrel and concrete are resisted by steel shear plates.
STRESS DISTRIBUTION AROUND A CIRCULAR HOLE IN A CIRCULAR CYLINDRICAL SHELL A survey was made of the available theoret ical solutions and experimental techniques for determining stress distributions around circular holes in a shell structure. This survey indicated that an available programmed finite-element solution was most practical for this application.
ANALYSIS OF STRESSES AROUND LARGE OPENINGS
A. Method of Analysis The adequacy of the design of the openings was verified by the use of Computer Program FELAP (Reference 12) developed at the Franklin Institute Research Laboratories (FIRL).
This program provides for the representation of the shell by flat rectangular panels with
multiple layers and is used on the assumption that the perturbation on the shell introduced
by the presence of the opening is local. The basis for the derivation of this program is
described in Reference 13.
B. Verification of Method Accuracy In order to evaluate the accuracy of the solution method, a test problem was solved to develop stresses which could be directly compared with other theoretical solutions and
experimental results. Very satisfactory results were obtained as evidenced by the data presented in Section 4.1. This study also verified the adequacy of the grids used on the
analysis of the openings for the Three Mile Island - Unit No. 1 Reactor Building.
C. Basis for Analytical Model For purposes of the analysis, the shell was idealized by representing (1) the liner as an isotropic steel layer, (2) the horizontal or vertic al reinforcement as an orthotropic layer with no Poisson's ratio effect and no shear stiffness, and (3) the circular ring reinforcement with
zero shear stiffness. A more complete description of the model idealization is contained in
Section 4.2.1.
TMI-1 UFSAR APPENDIX 5C 5C-2 REV. 18, APRIL 2006 D. Load Combinations
The basic loads, including dead, internal pressure, earthquake, prestress, and thermal (operating and accident) loads, were combined in accordance with the basic criteria and
more fully tabulated as shown in Table 1. All loading conditions are more fully tabulated as
shown in Table 1. All loading conditions are more fully described in Section 4.2.2.
E. Creep and Shrinkage Effects
The effect of load redistribution due to concrete shrinkage and creep was investigated as described in Section 4.2.3 and found to be negligible.
F. Summary of Results
The stress-resultants and stress-couples for panels which were of interest in evaluating the adequacy of the design are summarized in Tables 2 through 8. The correctness, of the
values computed by the finite-element method was verified in several ways, as described in
Section 4.1 (c).
VERIFICATION OF REINFORCEMENT ADEQUACY
A. General Method
Principal stress-resultants and stress-couples were computed for all panels. Interaction diagrams were prepared based upon procedures for ultimate strength design of ACI 318-63.
Interaction diagrams for those panels found to be significant
are included as Figures 20 through 28, indicating the stress state for all pertinent loading combinations. The interaction diagrams show that sufficient reinforcement has been provided to carry all loads, including the full thermal stress-resultants and stress-couples.
B. Additional Considerations
Additional studies were performed to evaluate the acceptability of the penetration barrel and reinforcement for normal shears, which are described in Sections 5.4 and 5.5.
C. Design Drawings
Design drawings providing details of the opening reinforcement are included at the end of this report.
TMI-1 UFSAR APPENDIX 5C 5C-3 REV. 18, APRIL 2006 1 DESIGN BASES
1.1 General
The large openings in the Reactor Building are designed generally in accordance with the criteria as detailed in the Safety Analysis R eport, and as more completely detailed hereafter.
The large openings consist of the opening for the equipment access hatch, with a diameter
of 22 feet 4 inches, and the opening for the personnel lock, with a diameter of 10 feet 0
inches Although these criteria and the analytical and design methods described hereafter
apply to both openings, detailed results are provided only for the larger opening.
1.2 Design
Loads
The following loads were considered in the structural design of the Containment Vessel:
- a. Internal Pressure-Design Pressure is 55 psig
- b. Test Pressure - 63.3 psig (1.15 times design pressure)
- c. Live Loads
- d. External pressure - 2.5 psig
- e. Wind Load
- f. Thermal Loads
- 1. Accident
- 2. Operating
- 3. Test
- g. Seismic Ground Accelerations
- h. Dead Load
- i. Prestressing Load
- j. Tornado Load
The thermal loads on the Reactor Building and their variation with time, are developed on the basis of the blow-down transients as detailed in the Safety Analysis Report. The
openings, as a portion of a Class I structure, are designed on the basis of a horizontal
ground acceleration of 0.06g as the design earthquake, and of 0.12g as the maximum
hypothetical earthquake. The vertical component is taken as 2/3 of the horizontal
component.
The equipment access hatch consists of a single door located outboard of the Reactor Building shell with a personnel lock inset in the door. The barrel (penetration sleeve) for the
hatch is consequently subjected to the accident pressure. The isolated personnel lock
consists of double doors, one located inboard and the other outboard of the Reactor
Building shell. The design considers the consequence of both doors being closed during the
accident as well as the extremely remote possibility that either one of the doors is opened
during the accident.
1.3 Load Combinations
The design is based upon limiting load factors which are used as the ratio by which accident, earthquake, and wind loads are multiplied for design purposes to ensure that the TMI-1 UFSAR APPENDIX 5C 5C-4 REV. 18, APRIL 2006 load deformation behavior is essentially elastic. The loads utilized to determine the required limiting capacity of any structural element are computed as follows:
- a. C=0.95D + 1.5P + 1.OT
- b. C=0.95D + 1.25P + 1.OT' + 1.25E
- c. C=0.95D + 1.OP + 1.OT
+ 1.OE' d. C=0.95D + 1.OW t + 1.OP t Symbols used in the above equations are defined as follows:
C: Required load capacity of section D: Deal load of structure P: Accident pressure load - 55 psig T: Thermal loads based upon temperature transient associated with 1.5 times accident pressure. T': Thermal loads based upon temperature transient associated with 1.25 times accident pressure.
T: Thermal loads based upon temperature transient associated with accident pressure. E: Seismic load based upon 0.06g ground acceleration.
E': Seismic load based upon 0.12g ground acceleration.
W t: Wind loads based on 300 mph tornado.
P t: Pressure load based on an internal pressure of 3 psig difference between inside and outside of the Reactor Building.
The acceleration response spectra and structural damping are as detailed in the Safety Analysis Report.
1.4 Material
Stress/Strain Criteria
- a. Concrete Reinforcement The deformed bars used for concrete reinforcement in the Reactor Building, are intermediate grade billet steel conforming to ASTM A615-68 Grade 40 with a
guaranteed minimum yield strength of 40,000 psi.
The design limit for tension members (i.e., the capacity required for factored loads) is based upon the yield stress of the reinforcing steel. No mild steel reinforcement is designed to experience strains beyond the yield point at the factored loads. The
load capacity so determined is reduced by a capacity reduction factor "f" which
provides for the possibility that small adverse variation in material strengths, workmanship, dimensions, and control, while individually within required tolerances and the limits of good practice, occasionally may combine to result in an
under-capacity. The coefficient "f" is 0.95 for tension, 0.90 for flexure, and 0.85 for
diagonal tension, bond and anchorage.
- b. Prestressing Tendons
The steel tendons for prestressing consist of 169 - 1/4 in. diameter wires using a BBRV anchorage system and high tensile, br ight, cold drawn, and stress-relieved steel wires conforming to ASTM A421-59T, Type BA, "Specifications for Uncoated TMI-1 UFSAR APPENDIX 5C 5C-5 REV. 18, APRIL 2006 Stress-Relieved Wire for Prestressed Conc rete" with a minimum ultimate tensile stress of 240,000 psi.
The steel tendons are stressed during the post-tensioning operation to a maximum of 80 percent of ultimate strength and locked-off for an initial stress of 70 percent of ultimate
strength. The maximum effective prestress is determined taking into consideration
allowances for the following losses, which are deducted from the transfer prestress:
- 1. Elastic shortening of concrete
- 2. Creep of concrete
- 3. Shrinkage of Concrete
- 4. Steel relaxation
- 5. Frictional loss due to intended or unintended curvature of the tendons
In no event does the effective prestress exc eed 60 percent of the ultimate strength of the prestressing steel, or 80 percent of the nominal yield point stress of the prestressing steel, whichever is smaller. The design is based upon the steel tendons not being stressed
beyond the yield point as defined by ACI 318-63 when subjected to the factored loads.
- c. Structural Concrete
The structural concrete will have a minimum compressive strength of 5,000 psi in 28 days. Under operating conditions the allowable concrete stresses are in accordance
with ACI 318-63 Part IV-A, "Structural Analysis and Proportioning of Members-Working Stress Design," and Part V, Chapter 26, "Prestressed Concrete."
The Reactor Building, including the large openings, is checked for the factored load combinations and compared with what is gener ally defined as the yield strength of the structure. For concrete, the yield strength is defined, except as described hereafter by the allowable stresses in ACI 318-63, Part IV-B, "Structural Analysis and Proportioning of
Members - Ultimate Strength Design." Concrete cracking is assumed when when the
principal tensile stress based upon all loads including thermal loads, exceeds 6f C or 424 psi.
- d. Liner
The liner was designed as participating with the concrete shell in carrying membrane forces (See 5.1). The stress limits established for the liner are consistent with those
limits for stress intensity (i.e., the difference between the algebraically largest
principal stress and the algebraically smallest principal stress at a given point)
defined in Section III of the ASME Boiler and Pressure Vessel Code and based upon
a working strength design consistent with that code.
The liner is carbon steel plate conforming to ASTM A283-67, grade C. This material has a minimum yield stress of 30,000 psi and a minimum ultimate tensile strength of 55,000 psi.
The liner is normally 3/8 inch thick in the cy lindrical portion of the structure. In the immediate vicinity of the large opening it is increased to a 1 1/8 inch thick reinforcing plate.
TMI-1 UFSAR APPENDIX 5C 5C-6 REV. 18, APRIL 2006 The barrel and reinforcing plate are carbon steel conforming to ASTM A516 Grade 60 Fire box Quality modified to ASTM A300. This mate rial has a minimum yield stress of 32,000 psi and a minimum ultimate tensile strength of 60,000 psi.
1.5 Test Condition
No specific stress or strain limits are established for the test condition. The factored load combinations previously described have been establis hed so as to ensure that the response due to design loads is essentially elastic. A check is made to ensure that no significant
permanent deformation of the structure occurs during the test. This means that following
the test there should be no visible permanent distortion of the liner and only small hairline
cracks should exist in the concrete.
1.6 Operating
Condition
The load combinations relevant to operating conditions are determined as reflected in Section 4.2.2. Allowable stresses are based upon those stipulated in ACI 318-63.
2 GENERAL DESCRIPTION OF OPENING REINFORCEMENT
2.1 Introduction
The normal flow of stresses in the hoop and meridional direction is obstructed by the large access holes in the Reactor Building. During normal operation of the plant, horizontal and
vertical prestress forces are the single largest stress contributor. These forces do not create
any difficulty in transferring horizontal and vertical compressive stresses around the
opening, but give the necessary reserve compressive stresses to counterbalance the
horizontal and vertical strains resulting from the accident pressure load. The horizontal and
vertical stresses in the reinforcing steel ar e very small during normal operation, but will theoretically increase up to almost yield stress due to the factored pressure load. These
hoop and meridional forces under pressure load will be transferred around the opening by
counter acting prestress forces and by hoop, meridional, and circular ring reinforcing steel.
2.2 Ring Reinforcement Splices
The circular ring reinforcing steel will be spliced by use of a Cadweld Rebar Splice. These splices are located at points of low rebar stresses, and no more than 1/3 of the ring steel will
be spliced in one section. This should eliminate any slip between Cadweld and rebar that
could occur at high rebar stresses.
Radial tensile stresses out from the center of the access opening are created due to the pressure load. Hoop and meridional reinforcement is provided to carry these stresses out from the opening and will be terminated where "pur e" membrane stresses exist in the wall.
2.3 Normal
Shear at Edge of Opening
The peripheral or normal shear reinforcement in the concrete around the Penetration Barrel is designed for the computed shear at a distance equal to or greater than, d/2 = 42.0 in.,
from the edge of the opening per ACI 318-63, or for twice the normal shear due to internal
pressure on the hatch, whichever is the larger of the two valves.
TMI-1 UFSAR APPENDIX 5C 5C-7 REV. 18, APRIL 2006
2.4 Prestressing
The maximum curvature for any horizontal or vertical tendon, around an opening, is at a radius of not less than 25.0 feet. When tendons are draped around penetrations in a
manner in which a radial force is exerted on the penetration, within the diameter of the
penetration, the spacing of the tendons are not less than 10 inches. The distance between
the penetration and the centerline of the first tendon for the equipment access and
personnel access penetrations is not less than 12 inches. The distance between tendons
when two layers are required is not less than 18 inches. The total angular change of the
tendons was laid out to keep frictional losses within a satisfactory margin, and to satisfy a
practical execution of the job in the field.
3 STRESS DISTRIBUTION AROUND A CIRCULAR HOLE IN A CIRCULAR CYLINDRICAL SHELL
3.1 Introduction
The first theoretical treatment of the problem presented by openings in thin shell structures is commonly attributed to Lur'e 1 , who obtained an approximate solution for the stress distribution around a very small circular cut-out in a thin circular cylindrical shell subjected to a homogeneous biaxial stress state [
x'[P. Lur'e derived the following expression for the edge stress:
Equation 3.1
[= ([x + [P) - 2([x + [P) cos2f
+ 3(1 - v
- 2) X d 2 [2[x + ([P - 3[x) cos2f] 16Rt
in which:
[x = meridional stress in the shell without the opening
[P = hoop stress in the shell without the opening f = angular coordinate (see Figure 2)
R = shell radius t = shell thickness d = diameter of cut-out = 2r
Equation (3.1) does not include bending effects, which can be quite important in thin shells.
Therefore, Lur'e's solution represented a relat ively minor improvement on the flat plate solution (Reference 2). In fact, for many years the design of reinforcement around shell openings was based on the classical stress concentration factors of flat plate theory. In this
area solutions for openings reinforced by means of a symmetrical circular doubler plate
were obtained by Sezawa and Kubo (Reference 3), Gurney (Reference 4), and Beskin (Reference 5). Solutions for the stress distribution around unreinforced and ring-reinforced
holes in flat plates can be found in Savin's (Refer ence 6) extensive treatment of the subject.
The plane stress approach is not satisfactory, however, for large openings. In this case large stress-couples may appear around the edge of the opening even when the shell elsewhere
is in a pure membrane state of stress. For tunately, very valuable results have recently TMI-1 UFSAR APPENDIX 5C 5C-8 REV. 18, APRIL 2006 become available. Withum (Reference 7) inv estigated the stress distribution in a cylinder weakened by a hole, subjected to torsion, by using a perturbation scheme. This technique
was extended by Kline 'et al'(Reference 8) to determine the stresses around a circular cut-out in a pressurized circular cylindrical vessel. This work, carried out at the General
Technology Corporation with the support of the Bureau of Ships of the U. S. Navy, was part
of a systematic theoretical investigation of two problems: (References 9 and 10).
- a. determination of the state of stress in the vicinity of a circular hole on a circular cylindrical shell subject to internal pressure.
- b. determination of the stress distribution in two normally intersecting cylindrical shells.
The solution of problem (a), for internal pressure as well as other practical loading conditions, has been presented by Naghdi and Eringen (reference 10). Lekkerkerker (Reference 11), extending Lur'e's approach, s olved the same problem for axial tension and torsion. He found excellent correlation bet ween his solution and experimentally determined stresses (using electric resistance strain gages) in a mild steel tube subject to torsion.
Stress concentration factors from References 8 to 11 are presentated in Figures 1 and 2.
The coefficients given may be directly applied to the calculation of maximum stresses at the
edge of the opening.
Unfortunately, no theoretical solution is available for reinforced openings or for non-isotropic shells (e.g., orthotropic reinforced concrete shells). The complexity of these problems is
such that they can only be dealt with by means of numerical or experimental methods.
Among the many possible approaches, the f inite-element method and the stress-freezing technique of three-dimensional photoelasticity have become especially attractive in recent years and are briefly evaluated in Sections 3.2 and 3.3.
3.2 Finite-Element Method
Steady progress in the finite-element approach has led to the possibility of determining the stress distribution around reinforced openings in shell structures with good accuracy. The
most important advantage of this method is its generality: reinforcing rings, variable shell
thickness, and material orthotropicity, for example, may be incorporated into the analysis
without difficulty. Unfortunately, the accuracy of the solution must still be evaluated by test:
For instance, results obtained using two different grids may provide an indication of
covergence. Alternatively, results obtained using a prescribed grid size and pattern may be
checked against some of the theoretical solutions of Section 3.1 or against experimental
values. The latter approach was followed in connection with the analysis of the stresses
around the openings for the Three Mile Island Reactor Building.
The designs of the reinforcement around the openings in the Three Mile Island Reactor Building were verified by using Computer Program FELAP (Reference 12), developed at
Franklin Institute Research Laboratories (FIRL). The solution is based on a representation
of the shell as an assembly of flat rectangular panels. In the first order shell theory
described by Zudans (Reference 13), which form ed the basis for the FIRL finite-element solution, the rotation about the normal to the shell middle surface is taken equal to zero.
Consequently, this leaves only five degrees-of-freedom associated with each nodal point. It
must be noted that the model with five degrees-of-freedom at a corner point, while "compatible" for plate problems, is unbalanced for shell problems in the third rotation (Reference 14). Although usually this unbalance does not affect the accuracy of the TMI-1 UFSAR APPENDIX 5C 5C-9 REV. 18, APRIL 2006 solution, it can lead to unrealistic results (Reference 14). Therefore, in applications to nuclear power plants, a verification of the results becomes desirable.
Connor and Brebbia (Reference 15) developed a sti ffness matrix for a thin shell element of rectangular plan and also noted that good results can be obtained with the finite-element
method using non-compatible displacement expansions, which do not include all the rigid
body displacements (Reference 15). Accor ding to Connor and Brebbia (Reference 15), it appears that curved elements lead to better results, for the same element size and shape, and are therefore more efficient than flat elements. However, comparison of results
obtained using two different types of curved triangular elements (Refences 16 and 17) with flat elements (References 12 and 14) for the test problem discussed in Section 4.1 did not
substantiate that belief (See Figure 10). In fact, all finite-element results were close to
Eringen, Naghdi, and Thiel's (Reference 9) solution, which can be considered, within the
limitations of shallow shell theory, an "exact" solution.
3.3 Applications
of Three-Dimensional Photoelasticity
Until very recently, experimental methods constituted the only feasible approach to the stress analysis of geometrically complicated reinf orced openings in shell structures. In this area, the stress-freezing technique of three-dimensional photoelasticity appears to be the
most suitable experimental method. Outstanding studies of stresses around reinforced
openings in pressure vessels were carried out by Levan (Reference 18), Taylor and Lind (Reference 19), and Takahashi and Mark (Reference 20). In the latter, comparison of the
photoelastic results with a finite-element analysis of the axisymmetric thick-walled reinforced concrete vessel showed very close agreement between the two solutions. Durelli, del Rio, Parks, and Feng (Reference 21) carried out an experimental evaluation of the stress around an opening in a thin shell by means of brittle coatings, electrical and mechanical strain
gages, micromenters, and photoelasticity with the objective of comparing the accuracy and
advantages of each technique.
Photoelasticity was concluded to be the most effective experimental approach in this type of problem (Reference 21). In the fabrication of the model Durelli 'et al' (Reference 21) used a
Hysol 4290 epoxy resin which was found to give poor performance in recent tests (Reference 22). Mark and Riera (Reference 22) believe that the use of the improved model materials will lead to a considerable reduction in the dispersion of photoelastic readings, which was large in the past (Reference 23), and which still is the most important argument
against this experimental approach. In spite of the difficulties associated with the model
material, the photoelasticity determined in Reference (21) shows good correlation with
Eringen and Naghdi's (Reference 9) theoretical solution. Stress concentration factors
determined photoelastically by Durelli 'e t al' and by Houghton and A. Rothwell (Reference 24) by means of electric resi stance strain gages mounted on circular cylindrical aluminum shells are given in Figure 3(c). Figure 3(c) also shows the stress concentration factors computed by Eringen and Naghdi (Reference 9).
The photoelastic approach presents several advantages (Reference 25): 1) full field observations give clear understanding of overall behavior and permits recognition of
unsuspected critical regions, 2) measurement s are made with very small effective gage lengths so that high gradients can be studied in small models, and 3) basic instrumentation
is simple and inexpensive. Traditionally, models have been machine finished, but improving casting techniques have already permitted the fabr ication of complicated models without any noticeable edge effect. This represents two important steps forward: 1) model fabrication TMI-1 UFSAR APPENDIX 5C 5C-10 REV. 18, APRIL 2006 cost can be drastically reduced and 2) the technique may now be applied to complicated models that cannot be readily machined.
Ducts for prestressing "tendons" have succe ssfully been incorporated into epoxy models by using nylon piano cords, which are set in the model pr ior to casting as if they were rebars in a conventional reinforced concrete element (Reference 26). After the epoxy has hardened
they can be easily pulled out, leaving a perfect duct without residual stresses around the
walls. The photoelastic method, therefore, offers the possibility of determining in one single
study the stresses around openings in prestressed vessels in "the large," as well as in "the
small." The latter, which includes stresses around curved tendons, corners, possible
non-linear distributions through the wall thickness, etc., may demand separate analyses
when a solution based on thin shell theory is used as the basis for design.
Further progress in theoretical or num erical analysis of stresses around shell openings, which must account for non-linear distributions through the shell thickness near the opening
and for other effects that cannot be predicted by thin-shell theory, can be fostered by
adequate experimental verification of the results, or by purely experimental studies that may help define the areas that require additional investigation. This is illustrated by a
photoelastic investigation of stresses around reinforced openings in plates due to
Lerchenthal (Reference 27), which indicates that the departure from a plane stress
distribution near the openings may be significant. For this purpose, the stress-freezing
technique may yet prove to be an invaluable tool.
TMI-1 UFSAR APPENDIX 5C 5C-11 REV. 18, APRIL 2006
- 4. ANALYSIS OF THE STRESSES AROUND LARGE OPENINGS IN THE THREE-MILE ISLAND UNIT NO. 1 REACTOR BUILDING
4.1 Verification
of Finite-Element Method of A nalysis as outlined in Section 3.1, a finite-element solution was chosen to determine stresses around large openings in the Reactor Building.
To evaluate the accuracy of the solution method (FIRL Program FELAP, as described in
Section 3.2), a test problem was solved with the same grid used in the finite-element
analysis of openings for the Three Mile Island Reactor Building. The grid is shown in Figure
- 4. In the finite-element analysis the following assumptions were made:
- a. The perturbation introduced by the presence of the opening on the shell state of stress is localized.
- b. According to (a.), stresses and displacem ents some distance away from the region surrounding the opening are not affected by the opening.
- c. A panel of rectangular plan, which is centered around the opening, is removed from the shell (Figure 4). The displacements corresponding to the shell without the
opening along the boundary lines of this panel constitute the boundary conditions for
the finite-element analysis. The analysis is correct if the panel boundaries are
sufficiently far from the opening so that (a.) and (b.) apply. This assumption can be
verified 'a posteriori' by comparing stresses along the boundary lines with those
existing in the shell without the opening.
- d. Each element can be idealized as a layered plate. Three typical elements, each with four layers, for representation of three regions of different thickness of the finite
element model, as shown in Figure 5.
- e. Because of symmetry, only one quadrant need be analyzed.
Since reliable experimental or theoretical result s for reinforced openings in shells similar to that under consideration were not available, it was decided to check the solution method
against the shell tested by Durelli 'et al' (Reference 21), for which other theoretical solutions
were also available. The example problem is defined by:
R = 430.00 in.
r = 85.70 in.
t = 18.04 in.
v = 0.30 p = 100.00 psi
The computed tangential membrane stresses around the opening edge are compared in Figure 6 with those given in Reference (9) and with the experimental stresses determined by Durelli 'et al' (Reference 21). Similarly, Figure 7 shows the tangential surface stresses
around the edge. Additional comparisons between the finite-element solution and the
experimental values are given in Figures 8 and
- 9. Finally, Figure 10 shows the variation along the symmetry axes of the stress resultant N P determined by different approaches, including another finite-element solution using triangular shell elements based on Prato's
work (Reference 16). The correlation of the FIRL finite-element solution with the other
results is good. It must be noted that the finite-element results are particularly close to the TMI-1 UFSAR APPENDIX 5C 5C-12 REV. 18, APRIL 2006 solution of Eringen 'et al' (Reference 9), which was regarded as the most accurate. It should also be pointed out that the correlation is good in spite of the fact that the panel boundaries
were not sufficiently removed from the opening, as revealed by Figure 9 which shows the existence of a small stress couple M P at s = 3.5 that had not been predicted by r
the model study. This result was verified by another finite-element analysis of the test problem using Prato's triangular shell elements (Reference 16).
It was pointed out in Section 3.1 that the stress concentration factors under loading conditions such as internal pressure, axial tension or torsion can be computed at different
locations in terms of the nondimensional parameter T (see Figure 2). As T 8 we approach
the plane stress solution and the convergence problem for the finite-element solution
disappears. It can be concluded, therefore, that satisfactory results in the test problem (T =
1.17) constitute adequate verification of the solution method for the large openings in the
Three Mile Island Reactor Building for which T = 62 (equipment hatch, based on typical shell
thickness).
4.2 Stress
Analysis Under Operating and Accident Loads
4.2.1 Representation
of the Shell Around the Opening
The finite-element grid used to solve the test problem (Section 4.1) was also employed to determine the stress distribution around the large openings in the Reactor Building. The
shell was idealized as follows:
- a. The liner was represented as an isotropic steel layer (E st = 30,000 ksi, U = 0.3).
Composite action was assumed in the determination of the stress resultants and
stress couples.
- b. The vertical and horizontal steel reinforcement was represented as a layer of an orthotropic material having no Poisson's ratio effect, no shear stiffness (G 12 = 8), and no meridional stiffness(E 1 = 8). This layer was located at the center of gravity of the meridional or hoop reinforcement.
- c. The ring steel rebars, providing additional reinforcement around the opening, were represented as a layer located at the center of gravity of the ring reinforcement. The shear stiffness of this layer was set to equal zero (G 12 = 8). In Computer Program FELAP the axes of orthotropicity are oriented in the hoop and meridional directions.
Consequently, since the axes of orthotropic ity of the ring reinforcement only coincide with those directions in the regions around the opening's vertical and horizontal axes, the following approximation was made:
at every panel type the hoop and meridional stiffnesses were directly proportioned to the projected steel area.
TMI-1 UFSAR APPENDIX 5C 5C-13 REV. 18, APRIL 2006
- d. The concrete layers were idealized as follows:
E 1 = 4000 ksi E 2 = 4000 ksi G 12 = 1740 ksi U 12 = 0.15
- e. The barrel (penetration sleeve) was represented on contributing, with 50 percent of its area, to the stiffness of the elements adjacent to the opening. In other words, a
stiffener of 3/8 inch thickness was included in the model along the periphery of the
opening.
Note that although the penetration barrel was incorporated into the finite-element model as explained above, it was not regar ded as contributing to the load-carrying capacity of the shell (See Section 5.2).
4.2.2 Basic
Loading Conditions
The stress distribution around the opening was determined for the loading conditions described in Section 1 and those loading combinations more completely described
hereafter. The specific loads are defined as follows:
- a. Dead Load
The stress distribution around the opening due to dead weight was calculated assuming a uniform meridional compression in the cylinder equal to the stress
resultant at the elevation of the opening axis (in the shell without the opening). The
weights of the equipment hatch and personnel lock were neglected. Since dead
weight stresses in the typical shell w all at the elevation of the equipment and personnel access are low (less than 100spsi), the above simplifications will not
significantly influence the final stresses in the pressurized vessel.
- b. Internal Pressure
Internal pressure was assumed to act on the interior of the shell as well as on the barrel (penetration sleeve) of the equipment access hatch. The internal pressure on
the hatch was assumed transferred to the shell by a uniform normal shear Q p around the opening. The effect of non-uniform distributions of Q p were analyzed separately.
- c. Earthquake
Seismic stresses around the openings were investigated for two directions of the horizontal component of motion:
- 1. Earthquake motion oriented in the direction normal to the openings.
- 2. Earthquake motion oriented at 900 with the direction normal to the openings.
Seismic loads were evaluated as described in Appendix B.
TMI-1 UFSAR APPENDIX 5C 5C-14 REV. 18, APRIL 2006
- d. Prestress
Prestressing loads are represented as two independent loading conditions by a uniform hoop and meridional compression in the shell (away from the opening) equal
to the stress resultant corresponding to initial and final prestress. Curved tendons
around the opening were considered in the analysis as line loads P = T/r where T is
the total prestressing force per tendon and r is the radius of curvature at the location
under consideration. These line loads were integrated within each panel and applied
as nodal forces on the shell, as shown in Figures 11 and 12.
- e. Thermal Loads
Operating Temperature
The steady-state temperature distribution in the reinforced area around the opening was obtained using "Heating 2" a computed code developed by Mr. R. D. Brunell of
Atomics International (Reference 28). This particular computer program is applicable for the solution of transient and/or steady state problems in one, two, or
three dimension cartesian or cy lindrical coordinate systems.
The steady state temperatures were determined using the formula:
M j K i (T i - T j) = 8 i=l
with convergence criterion
T j step - T j step A - 1 £ .00001 T j step A for all nodes.
The transient temperatures were determined using the formula:
M
j K i (T i - T j) = C j delta T j i=l delta T
to ensure stability of the transient solution the time step was chosen as:
delta T £ ( C j ) M j K i i=l
which was calculated for all nodes and the smallest value chosen.
The opening temperatures were determined for:
- 2. start-up in winter 3. shut-down in winter
- 4. accident condition in winter
The winter season was chosen because at this time the largest thermal gradients are present in the Reactor Building. The temperature on the external surface of the Reactor
Building was taken as 21.4F. This is the lowes t average monthly temperature ever recorded in the area by the United States Weather Bu reau. Lower temperatures will be experienced for shorter periods of time, however these only affect a small portion of the wall thickness, as proved by preliminary tests, and were not considered.
It was assumed that during start-up, the norm al operating temperature rose from 60 F to 110 F in one hour and for shut-down the reverse was used. The inside film coefficient was taken
as .010354 Btu/hr in 2 F for cases 1, 2, and 3 which is actually a modified film coefficient which takes into account the steel liner and the stagnant air gap between the liner and the
concrete:
h M = 1 X
STEEL X AIR 1
+
+ h i K STEEL K AIR
This does neglect any circumferential conduct ion, however, due to the small comparative thickness of the liner and air gap the effect will be negligible. This was also proven in
preliminary tests.
In case 4 the liner temperature was known therefore as a film coefficient of .040766 Btu/hr in 2 F was used which simply simulated the conduction through the air gap:
k AIR h M = X AIR The outside film coefficient in all cases was taken as .0069555 Btu/hr in 2 F.
The opening itself was broken into 61 nodes with adiabatic boundaries at the radial midpoint of the buttress and the radial midpoint of the plane wall. This was done because
considering symmetry, there is no heat transfer across these boundaries.
The following material values were used for the concrete:
P = .08102 lb/in.
3 C p = .210 Btu/lb F k = .06944 Btu/hr in F.
The input for the finite-element stress analysi s was prepared on the basis of the above results.
Summer thermal stresses are conservatively computed as equal to 27 percent of the peak winter stresses.
TMI-1 UFSAR APPENDIX 5C 5C-16 REV. 18, APRIL 2006
- f. Accident Temperature
The effect on the barrel (penetration) resulting from an accident temperature of 281F (net increase: 281 - 110 = 171F) was representated by internal pressure acting on the barrel.
The magnitude of this pressure was determ ined on the basis of a two dimensional analysis, which assumed the barrel (t = 0.75 in.) plus 0.55 in. of concrete were suddenly heated to
171 F. The equivalent pressure on the concrete was found to be equal to 183 psi.
Definitions for the foregoing variables are:
j K i = thermal conductivity between node j and i T i = temperature at node i T j = temperature at node J M = number of nodes adjacent to node j A = interation number
C j = compositance of node j T = time step X = thickness k = thermal conductivity
h M = modified film coefficient, inside h i = inside film coefficient h 8 = outside film coefficient P = density
C P = specific heat P 1 = radial distance between boundaries P 2 = radial distance opening covers
The temperature gradients obtained by t he aforemention computer program are presented as 6 figures (Figures 13 through 18). Figures 13, 14, and 15 refer to the
temperature gradients under winter operat ing temperature conditions at region near the opening, normal wall section, and the barrel-normal wall transition zone, respectively. Figures 16, 17, and 18 show the temperature gradients under winter accident temperature condition at region near the opening, normal wall section, and
the barrel-normal wall transition zone, respectively.
4.2.3 Effect
of Creep and Shrinkage
It appears that shrinkage of concrete can only introduce compressive stresses into the steel rebars. These stresses will largely disappear after the internal pressure in the shell takes
place, and need not be considered in the stress analysis of the opening.
The load redistribution due to concrete creep (i.e., the redistribution of Nf , NfP, NP , Mf, MP, etc.) is expected to be small. This conc lusion is sustained by a finite-element plane stress analysis, which indicates that the stress concentration factor for uniaxial compression
changes by only 5 percent when the hoop 'in-plane' stiffness is reduced by 100 percent. If the 'effective modulus' approach is used to determine the final stress distribution under
operating stresses, a final modulus equal to 65 percent of its initial value would lead to a
final ratio between vertical and hoop 'in-plane' stiffness equal to:
42 x 1600
= 67.3
= .950 41.875 x 1600 + .125 x 30,000 70.65 TMI-1 UFSAR APPENDIX 5C 5C-17 REV. 18, APRIL 2006 the initial ratio is:
42 x 4000
= 168
= .985 41.875 x 4000 + .125 x 30,000 170.8
Therefore, a change in the ratio between vertical and hoop 'in-plane' stiffnesses of about 4 percent is not expected to have any significant effect on the stress-resultant and
stress-couple distribution.
The effect of creep and shrinkage on prestress losses is taken into account as indicated in Section 5.8. Likewise, the transfer of load fr om concrete to steel in any given cross-section is considered in the verification of rebar stresses when applicable.
With reference to the discussion of Appendix A, it may be concluded that the load redistribution due to concrete creep in the unpressurized vessel will not affect in any
significant degree the load distribution in the structure under test or accident pressure.
- 5. VERIFICATION OF DESIGN CRITERIA
5.1 Basis
for Verification of Shell Loading Capacity Due to Primary Loads (Principal Stress-resultants and Principal Stress-couples)
The loading capacity at any point of the shell was verified according to the following procedure:
- a. The principal stress-resultants N 1 and N 2 were computed in terms of Nf, NP, and NfP.
- b. The principal stress-couples M 1 and M 2 were computed in terms of Mf, MP, and MfP.
- c. Considering that throughout the critical regions of the shell (both axes of symmetry and along the edge of the opening) the orientations of N 1 , N 2 , and M 1 , M 2 coincide with the orientation of the reinforcement and that in the rest of the shell they nearly coincide with each other and with the orientation of the steel rebars. Systems N 1 , M 1 , and N 2 , M 2 are treated independently. Since in the regions in which the directions of principal stress-resultants and stress-couples are not colinear or do not
coincide with the orientation of the steel rebars, stresses are low, the error
introduced by assuming them colinear will not affect the conclusion concerning the
load-carrying capacity of the shell. In the latter case, steel rebars not oriented in the
direction of principal stress-resultants or stress-couples were conservatively
neglected in the computation of the strength of the section. In other words, it is
proposed that forces in the 1-direction will not affect the strength of the section in the
2 direction and vice versa. This is in full agreement with a square failure criterion for
concrete in biaxial compression, which appears to be very conservative. Stresses in
rebars oriented along a principal stress direct ion obviously will not be affected by forces in the other principal direction.
TMI-1 UFSAR APPENDIX 5C 5C-18 REV. 18, APRIL 2006
- d. The computed ultimate capacity of any se ction of the shell satisfies the requirements of ACI 318-63, Sections 1600, 1700, 1800, and 1900. Only deformed bars as
defined in ACI 318-63, Sections 301 are used.
- e. Composite action between the shell and the liner is neglected in the computation of ultimate moments. The liner is regarded as carrying only its share of the principal stress-resultants.
Interaction diagrams were prepared as described in Section 5.2 for elements located along both symmetry axes and along a 45 degree line. Principal stress-resultants
and stress-couples corresponding to all critical load combinations are shown in
Figures 20 thru 28 on the interaction diagrams corresponding to elements 44, 55, 66, 77, 73, 74, 99, 100, and 101. The position of a point representing a stress state
within the diagram gives a clear indication of t he "local safety factor" at that location (i.e., at the center of the element). It should be pointed out that all points on the
interaction diagram are computed with respect to shell reference surface as shown in
Figures 20 thru 28. That is, under compression and bending, "N" and "M" were
transferred from the plastic centroid to the shell reference surface. Under tension, "N" was transferred from the center of gravity of the reinforcing steel to the shell
reference surface. It must be emphasized that even if a point representing a stress
state fell outside the diagram, that would not indicate a critical or nearly critical condition for two reasons:
- a. The interaction diagrams were determined on the basis of conservative assumptions and it is expected that the 'true' failure envelopes lie a certain
distance away from the computed envelopes.
- b. A point outside the interaction diagram would merely indicate local yielding of one or more rebars at that location, which would cause a load redistribution
towards less highly stressed regions. A point representing a stress state
contained within an interaction diagram indicates that stresses in all steel
rebars at that section are below yield stress, and that concrete stresses, where applicable, are below code allowable stresses.
5.2 Interaction
Diagram
- a. Axial Compression and Bending (See Figure 19)
- 1. Concentric Compression
P 8 = f [0.85 f C A C + A st f y],
A st = A s1 + A s2 + A s3
- 2. Simple Bending
a = A s1 f y + A s2 f s2 - A's3 f s3 0.85 f C b M 8 = [T 1 d 1 + T 2 d 2 + C s3 d'3 + C C (C - a)] 2 C d
TMI-1 UFSAR APPENDIX 5C 5C-19 REV. 18, APRIL 2006 M C M s1 + M C C C = 0.85 f C a b
- 3. Bending and Axial Compression
C b = d(87000) 87000 + f y a b = k 1 a.b P b = f [0.85 f C (a b - a) b]
M b = M 8 + P b d b
- 4. Determine P a and M a e a = 0.1 t; ACI 318-63, Section 1901 (a)
M a = P a e a By similar triangles, P a = P 8 M b e
a (P 8 - P b) + M b
- b. Axial Tension and Bending
- 1. Concentric Tension
P 8 = f A st f y Note: For purpose of clarity, only three steel layers
were included in the preceding equations.
5.3 Reinforcing
Steel
- a. The opening steel reinforcement will be as shown on Drawing Nos. D-421-035 and D-421-036.
- b. The amount of reinforcement, which includes regular hoop, meridional, and circular ring steel around the opening, equals or exceeds that shown to be required by
calculations. In no event is the liner assumed to contribute more than its yield
strength.
- c. The clear distance between parallel bars is not less than 2 times the maximum size of coarse aggregate, 2 times the bar diameter, nor less than 2 inches.
- d. Vertical shrinkage or temperature reinforcement is placed at the outside face of wall.
The minimum amount of such reinforcement on the outside concrete face wall is greater than 0.0015 of the gross cross-sectional area of concrete.
5.4 Penetration
Barrel
TMI-1 UFSAR APPENDIX 5C 5C-20 REV. 18, APRIL 2006 The portions of the Equipment Access Hatch and Personnel Lock extending beyond the concrete shell were designed and fabricated by Chicago Bridge and Iron Company in
accordance with the ASME Nuclear Vessels Code. The barrel of each of these penetrations
within the limit of the concrete thickness was investigated for the following loads:
- a. Meridional membrane stresses in the barrel due to internal pressure on the hatch.
- b. Hoop membrane stresses in the barrel due to the 'in-plane' deformation of the opening.
- c. Meridional bending stresses in the barrel caused by meridional shear transfer from the barrel to the concrete.
- d. Thermal Stresses.
The objective of this investigation was to verify that the stresses in the barrel are within allowable limits. Refer to ASME Nuclear Vessels Code, Article 4, Par. N-414.
It should be noted that the allowable stresses referred to are based on working
strength design.
5.5 Concrete
Shear
Splitting planes were hypothesized parallel to the surface of the shell through the various layers of concrete reinforcement and tendon conduit. The 'in-plane' shear stresses are
produced by radial forces produced by circular rebar rings and draped tendons. Sufficient
steel has been provided in the form of straight or hooked vertical and horizontal bars and
ties to develop the total shear stress across the hypothesized planes (Reference 29). The
shear stresses are conservatively assumed to be the summation of the loads resisted by the circular bars on the vertical axis due to the factored pressure load. That is to say, the
shearing force exerted across a plane through Layer 15 (see attached Drawing No.
D-421-036) is equal to the summation of rebar forces on Layers 15 and 16 on the vertical
axis. The maximum shear stress on the dowels due to the aforementioned load does not
exceed the yield stress of the dowels.
The dowels are anchored by mechanical anchorage (900 hooks) and/or sufficient bond development lengt h which is determined on the basis of Ultimate Strength Design provision of ACI 318-63. All rebars provided to resist the
aforementioned loads consist of A615 mater ial with a 40,000 psi minimum guaranteed yield strength.
5.6 Shear-Diagonal Tension
The ACI 318-63 recognizes the punching shear to be critical at a distance "d/2" out from the periphery for slabs and footings (See Sections 1207 and 1707). A beam type of diagonal tension failure is impossible due to the geometry and two directional stresses in the shell. We believe that a punching mode of failure is the type of
failure that should be and has been investigated. These shear stresses included in the
computer output take into account the effect of the pressure on the door plus the reinforced
area.
Two modes of shear transfer are considered. First, shear transfer through concrete without shear reinforcement. Second, disregarding the shear capacity of concrete, enough
reinforcing steel is provided to carry the normal shears by steel alone.
TMI-1 UFSAR APPENDIX 5C 5C-21 REV. 18, APRIL 2006 Ultimate peripheral or normal shear stress carried by concrete is computed by:
v u = Q d
v c = 4 f d (f C)1/2 (ACI 318-63, Section 1707) where:
Q = normal shear stress-resultant at the critical section
v u = nominal ultimate shear stress as a measure of diagonal tension
v c = allowable ultimate shear stress to be carried by concrete d = distance from extreme compression fiber to centroid of tension reinforcement f C = 28 days compressive strength of concrete f = 0.85
Shear reinforcement requirements
The ultimate shear capacity of the reinforcing steel alone is computed by:
v u = f [a sv f y] where:
A sv = cross-sectional area of reinforcing steel acting in tension across a potential diagonal tension crack
f y = yield strength of reinforcement f = 0.85 TMI-1 UFSAR APPENDIX 5C 5C-22 REV. 18, APRIL 2006
5.7 Shear
Transfer Plates
Steel shear transfer plates are provided at the intersection of the barrel penetration sleeve and liner, for both equipment and personnel accesses, as shown in Figure 29. The shear
plates, made of A516 Gr. 70 steel, are designed to resist a total shear force produced by an
internal pressure of 1.5 accident pressure acting outward from the barrel. The shear force is
transferred from the barrel penetration sleeve to the liner and concrete by means of the shear transfer plates. It is conservatively assumed that the welded tee joint, at the
intersection of the liner and barrel penetration, does not resist any of the shear force.
5.8 Radial
Reinforcement
Draped horizontal and vertical tendons produce radial forces which act toward the opening as shown in Figures 11 and 12. These radial forces result in producing tensile stresses, in
the concrete, in the vicinity of the curved tendons. Steel reinforcement is provided in the
form of hook and straight bars to resist radial forces, as can be seen in Drawing D-421-036.
Hoop reinforcing resist the radial forces pro- duced by curved vertical tendons while
meridional reinforcing resist radial forces produced by curved horizontal tendons.
The unit radial forces and total radial forces are given by the following equations:
Unit Radial Force p = T/r N
Total Radial Force p = Ti Li i=l ri
p = unit radial force kips/inch P = total radial force KIPS T = initial prestressing tendon force KIPS r = radius of curvature of tendon N = total number of curved tendons
5.9 Summary
of Design and Conclusions
In selecting the analytical method used for determining the stress-resultant and stress-couple distributions in the shell under all critical loading conditions an extensive
bibliographic search was conducted and available methods were evaluated. In our judgment
the analysis of the stresses around the openings for the Three Mile Island Reactor Building
has been based on the most satisfactory of available methods.
The design was guided by the basic proposition that the best reinforcement is in fact the least reinforcement that will satisfy the requirement for carrying the shell loads around the
opening and the normal shear applied along the opening edge into the shell out to a
distance from the opening until a membrane state of stress is reached. Although not directly
applicable the IITRI studies on steel containment stru ctures conclusively showed that a stiff reinforcement around openings substantially reduces the burst strength of circular plates (Reference 30). In our judgment this design as evidenced by the data included in Section 5
provides the required reinforcement strengt h and further conservatism in determining reinforcement requirements is not prudent in t hat the ultimate load capacity might be thereby reduced.
TMI-1 UFSAR APPENDIX 5C 5C-23 REV. 18, APRIL 2006 LIST OF REFERENCES
- 1. A. I. Lur'e: "Concentration of Stresses in the Vicinity of an Aperture in the Surface of a Circular Cylinder," Prikl. Mat. Mekh. 10 , 1946, pp. 397-407, (English translation by N.
Brunswick, N.Y. Univ., Inst. of Math. Science, 1960)
- 2. S. Timoshenko: "Strength of Materials , Part II," D. Van Nostrand Company, Inc., New York, N.Y., 1930, p. 457.
- 3. K. Sezawa and K. Kubo: "Stresses in a Plate with a Flanged Circular Hole," Report of the Aeronautical Research Institute , Tokyo Imperial University, Tokyo, Japan, No. 84, September, 1932.
- 4. G. Gurney: "Analysis of the Stresses in a Flate Plate with a Circular Hole Under Edge Forces," British Rep. Memo No. 1834 , February, 1938.
- 5. Leon Beskin: "Strengthening of Circular Holes in Plates Under Edge Loads," Journal of Applied Mechanics , Trans. ASME, Vol. 66, 1944, p. A-140.
- 6. G. N. Savin: "Stress Concentration Around Holes," Pergamon Press, 1961.
- 7. D. Withum: "Die Kreiszylinderschale mit Kreisformigem Ausschnitt unter Schubbean-Spruchung," Ingerieur Archiv, XXVI, 1958, pp. 435-446.
- 8. V. Kline, R. C. Dixon, N. F. Jordan, and A. C. Eringen: "Stresses in Pressurized Cylindrical Shell with Cut-out," General Technology Corp. Tech, Report 3-1 , 1961.
- 9. A. C. Eringen, A. K. Naghdi, and C. C. Thiel: "State of Stress in a Circular Cylindrical Shell with a Circular Hole," Welding Research Council Bulletin 102 , 1965.
- 10. A. K. Naghdi and A. C. Eringen: "Stress Distribution in a Circular Cylindrical Shell with a Circular Cut-out," Ingenieur Archiv.
- 11. J. G. Lekkerkerker: "Stress Concentration Around Circular Holes in Cylindrical Shells," Eleventh Int. Congress on Applied Mechanics , Munich, 1964.
- 12. FELAP - Finite Element Analysis Program for Complex Structures, Franklin Institute Research Laboratories , Philadelphia, Pa.
- 13. Z. Zudans: "New Formulation and Evaluation of Elastic Shell Theory
," Ph. D. Dissertation, University of Pennsylvania, May 1966.
- 14. Z. Zudans: "Analysis of Asymmetric Stiff ened Shell Type Structures by the Finite-Element Method," paper to be published in "Nuclear Engineering and Design
."
- 15. J. J. Connor and C. Brebbia: "Stiffness Matrix for Shallow Rectangular Shell Elements", ASCE Journal , Vol. 93, EM5, pp. 43-65, 1967.
TMI-1 UFSAR APPENDIX 5C 5C-24 REV. 18, APRIL 2006
- 16. C. Prato: "A Mixed Finite-Element Method for Shell Analysis
," Ph.D. Dissertation, M.I.T., June 1968.
- 17. F. Yao "Finite-Element Method for Shell Analysis," Ph.D. Dissertation, M.I.T., January 1968.
- 18. M. M. Leven: "Photoelastic Determination of Stresses in Reinforced Openings in Pressure Vessels," Welding Research Council Bulletin No. 113 , N.Y., 1966, pp. 25-52.
- 19. C. E. Taylor and N. C. Lind: "Photoelastic St udy of the Stresses Near Openings in Pressure Vessels," ibid., pp. 1-24.
- 20. S. K. Takahashi and R. Mark: "Analysis and Photoelastic Investigations of Stress Concentrations in Sphere-Cylinder Configurations
," U. S. Naval Civil Engineering Laboratory Report.
- 21. A. J. Durelli, D. J. del Rio, V.J. Parks, H. Feng: "Stresses in a Pressurized Cylinder with a Hole," Journal ASCE , Vol. 93, ST5, 1967, pp. 383-399.
- 22. R. Mark and J. D. Riera: Discussion of Ref. (19), Journal ASCE , Vol. 94, ST4, pp. 1075-76.
- 23. J. L. Mershon: "Preliminary Evaluation of PVCR Photoelastic Test Data on Reinforced Openings in Pressure Vessels," Welding Research Council Bulletin No. 113 , 1966.
- 24. D. S. Houghton and A Rothwell: "The Influence of Curvature on the State of Stress Ahead of Cracks in Cylindrical Shells," Non-Classical Shell Problems , Proc. of the Int. Symposium on Non-Classical Shell Problems, Warsaw, Poland, 1963, North Holland, 1964, pp. 685-700.
- 25. R. Mark: "Photomechanical Model Analysis of Concrete Structures," ACI Symposium on Models , 1968.
- 26. J. H. Barnwell and C. H. Edwards: "The Development and Testing of Three-Dimensional Photoelastic Models of Pretensioned, Prestressed Concrete Beams," paper presented at the
1968 SESA Spring Meeting , May 1968, Albany, N.Y.
- 27. Ch. H. Lerchenthal: "A Photoelastic Investigation of Stress Concentration Around Reinforced Cut-outs in Stresses Sheets," ibid.
- 28. R. D. Brunell "Computer Program Heating and Heating 2," Atomic International Division Report No. AI-64-MEMO-177, North American Aviation Inc., Detroit, Michigan, USA, 1964.
- 29. R. F. Mast: "Auxiliary Reinforcement in Concrete Connections," Journal ASCE , Vol. 94, No.
ST6, June 1968, pp. 1485-1504.
- 30. M. A. Salmon: "Studies of Reactor Containment Structures," IITRI Final Report for U.S.
AEC, Dept. of Commerce, Washington, D. C., 1966.
TMI-1 UFSAR APPENDIX 5C 5C-25 REV. 18, APRIL 2006
TABLES
TMI-1 UFSAR APPENDIX 5C 5C-1 REV. 18, APRIL 2006 FUNDAMENTAL LOADS TABLE 5C-1 Load No. DL (1) H&VP (2) OT (3) AT (4) IP (5) E 1 (6) E 2 (7) (8) 1 1.000 1.260 0.0 0.0 0.0 0.0 0.0 0.0 2 1.000 1.110 0.270 0.0 0.0 0.0 0.0 0.0 3 1.000 1.110 1.000 0.0 0.0 0.0 0.0 0.0 4 1.000 1.110 0.270 0.0 0.0 0.625 0.0 0.0 5 1.000 1.110 1.000 0.0 0.0 0.625 0.0 0.0 6 1.000 1.110 0.270 0.0 0.0 -0.625 0.0 0.0 7 1.000 1.110 1.000 0.0 0.0 -0.625 0.0 0.0 8 1.000 1.110 0.270 0.0 0.0 0.0 0.625 0.0 9 1.000 1.110 1.000 0.0 0.0 0.0 0.625 0.0 10 1.000 1.110 0.270 0.0 0.0 0.0 -0.625 0.0 11 1.000 1.110 1.000 0.0 0.0 0.0 -0.625 0.0 12 1.000 1.000 0.270 0.0 0.0 0.0 0.0 0.0 13 1.000 1.000 1.000 0.0 0.0 0.0 0.0 0.0 14 1.000 1.000 0.270 0.0 0.0 0.625 0.0 0.0 15 1.000 1.000 1.000 0.0 0.0 0.625 0.0 0.0 16 1.000 1.000 0.270 0.0 0.0 -0.625 0.0 0.0 17 1.000 1.000 1.000 0.0 0.0 -0.625 0.0 0.0 18 1.000 1.000 0.270 0.0 0.0 0.0 0.625 0.0 19 1.000 1.000 1.000 0.0 0.0 0.0 0.625 0.0 20 1.000 1.000 0.270 0.0 0.0 0.0 -0.625 0.0 Normal Operation 21 1.000 1.000 1.000 0.0 0.0 0.0 -0.625 0.0 22 1.000 1.110 0.270 0.0 1.150 0.0 0.0 0.0 23 1.000 1.110 1.000 0.0 1.150 0.0 0.0 0.0 24 1.000 1.000 0.270 0.0 1.150 0.0 0.0 0.0 Test Condition 25 1.000 1.000 1.000 0.0 1.150 0.0 0.0 0.0 26 1.000 1.110 0.270 1.000 1.000 0.0 0.0 0.0 27 1.000 1.110 1.000 1.000 1.000 0.0 0.0 0.0 28 1.000 1.110 0.270 1.000 1.000 1.000 0.0 0.0 29 1.000 1.110 1.000 1.000 1.000 1.000 0.0 0.0 30 1.000 1.110 0.270 1.000 1.000 -1.000 0.0 0.0 31 1.000 1.110 1.000 1.000 1.000 -1.000 0.0 0.0 32 1.000 1.110 0.270 1.000 1.000 0.0 1.000 0.0 33 1.000 1.110 1.000 1.000 1.000 0.0 1.000 0.0 34 1.000 1.110 0.270 1.000 1.000 0.0 -1.000 0.0 35 1.000 1.110 1.000 1.000 1.000 0.0 -1.000 0.0 36 1.000 1.000 0.270 1.000 1.000 0.0 0.0 0.0 37 1.000 1.000 1.000 1.000 1.000 0.0 0.0 0.0 38 1.000 1.000 0.270 1.000 1.000 1.000 0.0 0.0 39 1.000 1.000 1.000 1.000 1.000 1.000 0.0 0.0 40 1.000 1.000 0.270 1.000 1.000 -1.000 0.0 0.0 Loading Combinations Accident Pressure Loadings 41 1.000 1.000 1.000 1.000 1.000 -1.000 0.0 0.0 TMI-1 UFSAR APPENDIX 5C 5C-2 REV. 18, APRIL 2006 FUNDAMENTAL LOADS (continued)
TABLE 5C-1 (continued)
Load No. DL (1) H&VP (2) OT (3) AT (4) IP (5) E 1 (6) E 2 (7) (8) 42 1.000 1.000 0.270 1.000 1.000 0.0 1.000 0.0 43 1.000 1.000 1.000 1.000 1.000 0.0 1.000 0.0 44 1.000 1.000 0.270 1.000 1.000 0.0 -1.000 0.0 45 1.000 1.000 1.000 1.000 1.000 0.0 -1.000 0.0 46 1.000 1.110 0.270 1.000 1.250 0.0 0.0 0.0 47 1.000 1.110 1.000 1.000 1.250 0.0 0.0 0.0 48 1.000 1.110 0.270 1.000 1.250 0.625 0.0 0.0 49 1.000 1.110 1.000 1.000 1.250 0.625 0.0 0.0 50 1.000 1.110 0.270 1.000 1.250 -0.625 0.0 0.0 51 1.000 1.110 1.000 1.000 1.250 -0.625 0.0 0.0 52 1.000 1.110 0.270 1.000 1.250 0.0 0.625 0.0 53 1.000 1.110 1.000 1.000 1.250 0.0 0.625 0.0 54 1.000 1.110 0.270 1.000 1.250 0.0 -0.625 0.0 55 1.000 1.110 1.000 1.000 1.250 0.0 -0.625 0.0 56 1.000 1.000 0.270 1.000 1.250 0.0 0.0 0.0 57 1.000 1.000 1.000 1.000 1.250 0.0 0.0 0.0 58 1.000 1.000 0.270 1.000 1.250 0.625 0.0 0.0 59 1.000 1.000 1.000 1.000 1.250 0.625 0.0 0.0 60 1.000 1.000 0.270 1.000 1.250 -0.625 0.0 0.0 61 1.000 1.000 1.000 1.000 1.250 -0.625 0.0 0.0 62 1.000 1.000 0.270 1.000 1.250 0.0 0.625 0.0 63 1.000 1.000 1.000 1.000 1.250 0.0 0.625 0.0 64 1.000 1.000 0.270 1.000 1.250 0.0 -0.625 0.0 65 1.000 1.000 1.000 1.000 1.250 0.0 -0.625 0.0 66 1.000 1.000 0.270 1.000 1.150 0.0 0.0 0.0 67 1.000 1.260 1.000 1.000 1.150 0.0 0.0 0.0 68 1.000 1.000 0.270 1.000 1.150 0.0 0.0 0.0 Accident Pressure Loadings (continued) 69 1.000 1.000 1.000 1.000 1.150 0.0 0.0 0.0 70 1.000 1.110 0.270 0.0 0.0 1.000 0.0 0.0 71 1.000 1.110 1.000 0.0 0.0 1.000 0.0 0.0 72 1.000 1.110 0.270 0.0 0.0 -1.000 0.0 0.0 73 1.000 1.110 1.000 0.0 0.0 -1.000 0.0 0.0 74 1.000 1.110 0.270 0.0 0.0 0.0 1.000 0.0 75 1.000 1.110 1.000 0.0 0.0 0.0 1.000 0.0 76 1.000 1.110 0.270 0.0 0.0 0.0 -1.000 0.0 Loading Combinations (continued)
Normal Operations 77 1.000 1.110 1.000 0.0 0.0 0.0 -1.000 0.0 DL = Dead Load AT = Accident Temperature Note: Coefficients are based on: H & VP = Horizontal and Vertical Prestress IP = Internal Pressure Accident Pressure, Winter Temperature, OT = Operating Temperature E1 = Earthquake 1 Final Prestress, Maximum Hypothetical Earthquake E2 = Earthquake 2 TMI-1 UFSAR APPENDIX 5C 5C-1 REV. 18, APRIL 2006 TABLE 5C-2 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 1 DEAD LOAD MEMBRANE TWISTING AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q 11 -5.55 14.78 -0.15 3.58 0.12 -0.36 0.42 0.00
22 -5.05 -17.67 -1.05 -0.60 0.21 -0.25 0.46 -0.01
33 -4.61 -59.53 -0.94 -10.57 0.43 5.09 0.27 0.26
44 -2.06 -8.70 1.36 25.65 0.79 15.23 -1.26 0.10
55 -0.50 8.10 4.01 77.44 0.71 10.54 -0.66 -0.54
66 0.30 13.15 7.17 139.22 0.45 3.47 -0.48 -1.01
77 0.44 4.67 12.26 241.50 -0.63 -16.09 0.65 -0.88
25 -8.70 -22.61 0.78 3.48 -0.32 -9. 85 0.23 0.08
49 -12.24 -210.41 2.13 64.37 -0.28 -16.70 0.82 -0.53
73 -14.62 -259.86 1.26 48.47 0.30 -10.33 0.86 -0.25
74 -11.32 -198.03 0.58 10.96 2.43 31.48 0.42 0.86
94 -8.11 11.44 0.79 -8.53 -0.12 8.44 -2.32 -0.12
97 -11.96 -142.28 -1.19 5.85 -0.30 5.98 0.03 0.43
99 -16.70 -279.92 -3.20 -45.84 -0.38 -10.72 0.11 1.25
100 -21.34 -366.85 -3.39 -56.70 0.28 0.89 1.12 1.51
101 -29.35 -504.98 -1.93 -11.53 2.56 48.08 2.72 -2.36
NOTE: N, N, N, Q, and Q in Kips/in.
M and M in Kips-in/in. Values computed by finite-element analysis (Sheet 1 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-2 REV. 18, APRIL 2006 TABLE 5C-3 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 2 HOOP AND VERTICAL PRESTRESS MEMBRANE TWISTING AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q
11 -31.79 -142.30 -72.32 55.63 0.14 -1.20 -0.44 0.12
22 -29.51 -118.97 -67.36 61.36 0.73 -2.52 -0.35 0.05
33 -24.90 -167.75 -91.91 -789.23 2.69 12.76 -0.50 32.16
44 -8.42 2.42 -113.30 -1906.16 6.02 64.67 -3.67 2.62
55 -1.13 -13.56 -112.86 - 1752.00 9.22 107.44 -10.39 2.84
66 -9.01 -80.17 -118.33 -1631.09 10.90 122.42 -18.66 0.59
77 -10.19 -17.42 -125.59 -1570.90 14.17 124.75 -27.16 3.44
25 -39.83 -117.50 -74.70 -241.10 1.72 -150.89 2.59 4.00
49 -45.42 -379.25 -69.20 -751.08 16.13 33.18 0.22 3.20
73 -52.07 -458.64 -53.36 -659.38 29 .47 224.47 0.31 2.26
74 -53.81 -488.44 -60.14 -809.52 42.35 392.40 10.25 6.37
94 -39.10 47.94 -55.55 9.88 0.52 -4.06 0.73 3.52
97 -60.31 -475.61 40.43 -255.04 2.58 22.90 -0.42 -4.61
99 -68.06 -543.16 -17.65 -86.70 7.30 84.43 2.09 -6.53
100 -72.22 -416.40 -6.67 -30.10 9.89 93.83 -1.10 10.62
101 -76.92 -219.28 -7.43 26.58 10.27 46.46 -4.02 14.42
NOTE: N, N, N, Q, and Q in Kips/in.
M and M in Kips-in/in.
Values computed by finite-element analysis (Sheet 2 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-3 REV. 18, APRIL 2006 TABLE 5C-4 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 3 OPERATING TEMPERATURE (WINTER)
MEMBRANE TWISTING AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q
11 -3.10 514.41 10.23 433.38 -0.17 0.10 0.20 -0.01
22 -3.96 529.61 7.96 440.66 -0.12 -0.61 -0.69 -0.02
33 0.24 694.27 0.39 984.76 0.11 -7.74 -5.00 -33.98
44 -4.29 490.90 -14.95 1582.80 0.27 -38.15 -1.04 0.08
55 -4.07 323.86 -18.51 1652.76 0.08 -97.75 1.43 -3.91
66 -3.40 268.22 -22.05 1771.96 0.25 -132.39 3.86 1.59
77 -1.16 145.44 -25.85 1932.96 2.60 -156.72 3.68 12.47
25 0.16 514.79 2.91 571.08 -3.92 101.63 -1.85 -2.50
49 -5.83 1144.06 -11.13 759.89 1.09 -38.77 -1.25 0.41
73 -6.41 1346.10 -6.66 730.25 4.13 -294.50 5.43 0.54
74 -5.76 1196.31 -7.02 955.61 4.50 -606.81 -3.38 -3.62
94 -17.97 489.90 -6.49 479.64 -4.98 102.04 -24.95 -0.16
97 -13.53 924.93 -4.45 642.63 0.50 -2.91 0.46 9.88
99 -8.48 1683.20 -3.26 478.10 0.57 -51.10 0.58 -1.90
100 -9.16 1866.96 -2.10 341.98 0.68 -106.07 0.01 -3.47
101 -12.72 2053.34 -0.91 35.57 1.61 -187.64 -0.52 7.54
NOTE: N, N, N, Q, and Q in Kips/in. M and M in Kips-in/in. Values computed by finite-element analysis (Sheet 3 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-4 REV. 18, APRIL 2006 TABLE 5C-5 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 4 ACCIDENT TEMPERATURE (WITH BARRELL EFFECT)
MEMBRANE TWISTING AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q 11 -2.86 217.29 0.09 186.63 -0.17 0.73 0.74 0.00
22 -3.74 170.86 0.68 176.19 -0.24 0.58 0.70 0.00
33 -4.34 140.97 4.97 333.57 -0.62 -8.70 0.40 -11.43
44 -9.82 -78.19 13.42 666.08 -1.80 -44.84 2.41 -0.05
55 -14.81 -278.23 17.75 787.17 -3.77 -108.26 3.15 -0.94
66 -20.71 -431.10 15.33 785.34 -6.40 -154.00 1.21 1.63
77 -27.65 -638.49 17.94 927.02 -10.08 -217.19 -2.52 18.04
25 -1.56 179.23 -1.02 205.89 -0.29 20.06 -0.44 -0.51
49 2.31 357.30 0.12 260.03 -4.03 -174.35 -1.44 -0.38
73 7.88 563.30 -0.85 185.09 -10.00 -353.01 -2.05 -0.07
74 0.99 356.46 0.62 274.74 -15.13 -534.57 0.72 0.32
94 -1.48 182.56 -1.67 168.19 -0.03 -0.75 0.08 -0.45
97 5.09 435.56 -4.30 164.76 -0.68 -17.07 -0.28 3.49
99 13.85 856.12 -12.86 129.17 -2.98 -87.04 -0.45 3.00
100 15.84 956.25 -20.01 -343.88 -6.20 -164.51 -0.64 5.65
101 7.86 892.45 -26.94 -613.86 -9.40 -255.04 16.46 -2.69
NOTE: N, N, N, Q, and Q in Kips/in.
M and M in Kips-in/in. Values computed by finite-element analysis (Sheet 4 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-5 REV. 18, APRIL 2006 TABLE 5C-6 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 5 55 psig INTERNAL PRESSURE
MEMBRANE TWISTING AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q 11 23.12 88.28 45.64 -37.45 -0.12 1.52 0.81 -0.04
22 22.51 35.40 43.89 -56.46 -0.13 2.98 1.00 -0.02
33 22.43 29.84 58.58 423.42 -0.39 6.13 -0.18 -18.10
44 18.98 -5.22 74.08 1070.76 -1.22 4.56 0.85 -0.06
55 14.66 -55.50 82.16 1157.09 -3.64 -54.47 6.79 -3.58
66 8.84 -90.60 93.92 1304.67 -6.74 -96.68 8.83 -3.47
77 -1.05 -194.66 111.53 1589.83 -12.51 -160.33 13.96 2.52
25 20.98 52.28 47.42 169.43 -0.63 78.06 -1.62 -2.49
49 23.20 115.32 47.20 639.42 -10.54 -98.54 -1.12 -2.14
73 25.75 267.38 36.81 532.50 -20.32 -324.05 -3.19 0.21
74 25.24 213.57 39.74 607.13 -28.21 -427.25 -12.22 -0.96
94 20.55 -38.89 39.23 -110.38 -0.20 -7.25 1.36 -3.85
97 34.65 531.60 29.36 346.81 -1.70 -21.80 -0.58 5.64
99 36.80 701.25 13.72 103.07 -4.49 -81.21 -1.10 5.31
100 34.78 625.07 3.72 -34.14 -6.12 -109.36 -0.53 7.93
101 29.14 475.54 -5.16 -172.79 -6.51 -115.10 6.09 6.90
NOTE: N, N, N, Q, and Q in Kips/in.
M and M in Kips-in/in.
Values computed by finite-element analysis (Sheet 5 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-6 REV. 18, APRIL 2006 TABLE 5C-7 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 6 EARTHQUAKE #1
MEMBRANE TWISTING AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q 11 -4.07 10.83 -0.11 2.62 0.09 -0.26 0.31 0.00
22 -3.70 -12.95 -0.77 -0.44 0.15 -0.18 0.34 -0.01
33 -3.38 -43.64 -0.69 -7.75 0.32 3.73 0.20 0.19
44 -1.51 -6.38 1.00 18.80 0.58 11.16 -0.92 0.07
55 -0.37 5.94 2.93 56.76 0.52 7.73 -0.48 -0.40
66 0.22 9.64 5.26 102.05 0.33 2.54 -0.35 -0.74
77 0.32 3.42 8.99 177.02 -0.46 -11.79 0.48 -0.65
25 -6.38 -16.57 0.57 2.55 -0.23 -7.22 0.17 0.06
49 -8.97 -154.23 1.56 47.18 -0.21 -12.24 0.60 -0.39
73 -10.72 -190.48 0.92 35.53 0.22 -7.57 0.63 -0.18
74 -8.30 -145.16 0.43 8.03 1.78 23.07 0.31 0.63
94 -5.94 8.39 0.58 -6.25 -0.09 6.19 -1.70 -0.09
97 -8.77 -104.29 -0.87 4.29 -0.21 -4.38 0.02 0.32
99 -12.24 -205.18 -2.35 -33.60 -0.29 -7.86 0.08 0.92
100 -15.64 -268.90 -2.48 -41.56 0.21 0.65 0.82 1.11
101 -21.51 -370.15 -1.41 -8.45 1.88 35.24 1.99 -1.73
NOTE: N, N, N, Q, and Q in Kips/in.
M and M in Kips-in/in.
Values computed by finite-element analysis (Sheet 6 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-7 REV. 18, APRIL 2006 TABLE 5C-8 (Sheet 1 of 1)
STRESS AROUND EQUIPMENT HATCH LOADING CONDITION NO. 7 EARTHQUAKE #2 MEMBRANE TWISTING
AXIAL DIRECTION HOOP DIRECTION SHEAR MOMENT NORMAL SHEARS N M N M N M Q Q 11 -0.56 1.48 -0.02 0.36 3.59 -0.04 0.04 0.00
22 -0.51 -1.77 0.69 -0.06 2.72 -0.03 0.05 0.00
33 -0.46 -5.95 1.41 -1.06 2.04 0.51 0.03 0.03
44 -0.21 -0.87 2.14 2.57 1.58 1.52 -0.13 0.01
55 -0.05 0.81 2.90 7.74 1.07 1.05 -0.07 -0.05
66 0.03 1.32 3.72 13.92 0.55 0.35 -0.05 -0.10
77 0.04 0.47 4.81 24.15 -0.06 -1.61 0.07 -0.09
25 -0.87 -2.26 0.08 0.35 3.55 -0.99 0.02 0.01
49 -1.22 -21.04 0.21 6.44 3.55 -1.67 0.08 -0.05
73 5.54 -25.99 7.13 4.85 7.03 -1.03 0.09 -0.03
74 7.77 -19.80 8.96 1.10 9.14 3.15 0.04 0.09
94 -0.81 1.14 0.08 -0.85 3.57 0.84 -0.23 -0.01
97 -0.20 -14.23 -0.12 0.59 2.47 -0.60 0.00 0.04
99 0.13 -27.99 -0.32 -4.58 1.76 -1.07 0.01 0.13
100 0.37 -36.69 -0.34 -5.67 1.03 0.09 0.11 0.15
101 0.64 -50.50 -0.19 -1.15 0.26 4.81 0.27 -0.24
NOTE: N, N, N, Q, and Q in Kips/in.
M and M in Kips-in/in.
Values computed by finite-element analysis (Sheet 7 of 7)
TMI-1 UFSAR APPENDIX 5C 5C-8 REV. 18, APRIL 2006
FIGURES
TMI-1 UFSAR APPENDIX 5C 5C-9 REV. 18, APRIL 2006
DRAWINGS
TMI-1 UFSAR APPENDIX 5C 5C-10 REV. 18, APRIL 2006
APPENDIX A Effect of Concrete Creep and the Sustained Operating Stresses on Stress Distribution Around Openings in a Rapidly Pressurized Reinforced Concrete Vessel.
TMI-1 UFSAR APPENDIX 5C 5C-11 REV. 18, APRIL 2006 APPENDIX A EFFECT OF CONCRETE CREEP AND THE SUSTAINED OPERATING STRESSES ON THE STRESS DISTRIBUTION AROUND OPENINGS IN A RAPIDLY PRESSURIZED REINFORCED CONCRETE VESSEL
Consider the simple structure shown below:
Figure A-1
Column 1 is a reinforced concrete column with net concrete area A 1 C and longitudinal steel areas A st 1. A 2 C and A 2 st denote the net concrete area and the logitudinal steel area, respectively, of reinforced concrete columns 2. The system is loaded at time t = 0 with a vertical load P. Let us
determine the initial load distribution:
P = T 1 + 2T 2 sin 600 = T 1 + 3 T 2 (1)
delta L 2 = delta L 1 sin 600; delta L 2 = 3 delta L 1 (2) 2
in which L 1= T 1 (3) L A 1 C E C + A st E st 1 3 delta L 2 = T 2 (4) 2 L A
C 2 E C + A st 2 E st E C and E st denote the "effective" modulae of elasticity of concrete and steel, T 1 and T 2 the loads carried by columns 1 and 2, respectively. From equations (1) to (4) we obtain:
T 1 + 3 T 2 = P 3 delta L T 1 - 2 L T 2 = 0 2 1 1 3 A C E C + A st E st A C 2 E C + A 2 st E st
or T 1 J 1.5I + J
= P
{ { 3 I
} T 2 2 1.5I + J (5)
TMI-1 UFSAR APPENDIX 5C 5C-12 REV. 18, APRIL 2006 In which I = L
A C 1 E C + A st 1 E st J = 2 L
3 A
C 2 E C + A st 2 E st (6)
Equation (5) gives the loads acting on Columns 1 and 2 in terms of the "effective" modulae E C and E st In general, T 1 and T 2 will change with time (Load redistribution) due to concrete creep. Under the assumption that concrete behaves like a Kelvin-type material, the load distribution may be
calculated exactly for t ¥ by resorting to the creep-limit modulus E cu. (Time-dependent behavior of steel is neglected).
Note that even when there is no load redistribution (for example, when I = B) there will be some stress redistribution. In other words, T 1 an T 2 may remain constant, but the percentage of both carried by the steel reinforcement will increase as concrete creeps. As a result, steel stresses will increase and concrete stresses will decrease to final values which may be easily computed.
Let us now assume that at time t 1 a load P 1 issuperimposed on the existing load P 8 giving a total load (P 8 + P 1). (See Figure A-1). let T 1 and T 2 be the column loads immediately before the load P 1 is applied. To compute the actual column loads T 1 and T 2 after P 1 is applied, we would be tempted to determine the column loads T 1* and T 2* corresponding to P 1 acting alone on the structure (On the basis of the initial modulus of elasticity) and then add them to T 1 1 and T 2 1:
T 1 2 = T 1 1 + T 1* (7)
T 2 2 = T 2 1 + T 2* (8)
The approach is valid if there is no concrete cracking. If either column 1 or 2 (or both) cracks due to the resulting tensile stresses, the results obtained by restoring to equation (7) will be incorrect. The
situation will be best illustrated by an example. Let:
E C o = initial modulus of elasticity of concrete = 4000 ksi
E u C = "final" effective modulus of elasticity of concrete = 2000 ksi
E st = 7.5E C o = 30000 ksi
A C 1 = 100 in.
2 TMI-1 UFSAR APPENDIX 5C 5C-13 REV. 18, APRIL 2006 Ac 2 = 100 in.
2 A st 1 = 2 in.2 A st 2 = 4 in.2 L = 1000 in.
A load P 8 = -200 kips is applied at T = 8 and kept constant.
At time t = t 1 a second load P 1 = +300 kips is applied.
Initial Load Distribution under P = P 8
i = 1000 1 = 8.69 100 + 2 x 7.5 E C 8 E C 8 (9) J i = 2 1000 1 = 8.875 3 100 + 4 x 7.5 E C 8 E C 8 (10) T 1 8 = 8.875 P 8 = 0.404P 8 1.5 x 8.69 + 8.875 T 2 8 = 8.69 x 0.866 P 8 = 0.343P 8 (11) 1.5 x 8.69 + 8.875
Load Distribution under P = P 8 Immediately Before Application
of P 1 (t 1 ¥)
f = 1000 1 = 15.4 0.5 x 100 + 2 x 7.5 E C 8 E C 8 (12)
J f = 1.155 x 1000 1 = 14.45 0.5 x100 + 4 x 7.5 E C 8 E C 8 1 T 11 = 14.45 P 8 = 0.385P 1.5 x 15.4 + 14.45
1 T 2 = 15.4 x 0.866 P 8 = 0.355P 8 1.5 x 15.4 + 14.45 (13)
Concrete Stresses Due to P = P 8
8 f
C1 = Ii T 1E 8 e = -704 psi Initial Stresses L
{
8 8 f C2 = Ji T 2 E C = -608 psi L (14)
TMI-1 UFSAR APPENDIX 5C 5C-14 REV. 18, APRIL 2006 1-u f C1 = If T l E C = -598 psi Final Stresses L
(t 1 ¥ ) { 1-u f C2 = Jf T 2E C = -514 psi L (15)
Consequently, if t 1 is large, concrete stresses immediately after P 1 is applied would be:
f C 1+1 = -598 + 1055 = +457 psi
f C 1+2 =-514 + 912 = +398 psi (16)
If the tensile strength of concrete is f t = 420 psi, then column 1 should be expected to crack, i.e., the entire load T 1 would be carried by the steel reinforcement. Under such conditions, a load redistribution would occur, resulting in a large increase in T 2 , with subsequent cracking of columns 2 as well. The final load distribution will therefore depend on A st 1 and A st 2 only. That is to say, if both columns crack it is irrelevant whether t 1 is small or large. Moreover, A C 1 and A C 2 will no longer play any role in the problem. In fact:
I 1+ = 1000
= 0.01667 2 x 30000
J 1+ = 1.155 x 1000
= 0.00962 (17) 4 x 30000
T 1 1+ = 0.00962 P
= 0.278P 1.5 x 0.01667 + 0.00962
T 2 1+ = 0.01667 x 0.866
= 0.416P 1.5 x 0.01667 + 0.00962 (18)
It has been shown that if our sample structure is fully cracked, then creep has no influence
whatsoever on the final load distribution. It may be hypothesized, however, that the concrete
tensile strength in Column 2 is higher, say f t = 1500 psi. The question is than asked, what is now the load distribution. The problem may be solved by computing the total load "C" carried by concrete in Column 1 at time shortly before t = t 1 , and assuming that, as Column 1 cracks, load will be transferred simultaneously to joint and to the steel of column 1.
Initial Load Distribution with Concrete in Column 1 Fully Cracked
I c = 1000 1 = 66.6 2 x 7.5 E C 8 E C 8 J 8 = 1.155 x 1000 8.875 100 + 4 x 7.5 E C 8 (19)
T 1 = 8.875 P = 0.082 P 1.5 x 66.6 + 8.875
TMI-1 UFSAR APPENDIX 5C 5C-15 REV. 18, APRIL 2006 T 2 = 0.866 x 66.6 P = 0.531 P 1.5 x 66.6 + 8.875 (20)
T 1 = 0.385 P 8 + 0.082 (P 1 + C) - C
T 2 = 0.355 P 8 + 0.531 (P 1 + C) (21)
with C = -.598 x 100 = -59.8 kips we get:
T 1 = +2.6 kips t 1 ¥ T 2 = +56.9 kips } (22)
If t 1 is sufficiently small, it may be assumed that P 8 and P 1 are applied simultaneously at t = 0, in which case:
T 1 = 0.082 x 100 = +8.2 kips t 1 = 0 T 2 = 0.531 x 100 = +53.1 kips } (23)
The difference between (22) and (23) is not large. Note that superimposing the load distribution
after creep due to P 8 (equation 13) with the load distribution corresponding to the cracked structure under P 1 leads to:
T 1 =-0.385 x 200 + 0.082 x 300 = - 42.4 kips T 2 =-0.355 x 200 + 0.531 x 300 = 88.3 kips (24)
which are entirely unrealistic figures. Note also t hat an "exact" stress analysis for the case when t 1 is large (equations (19) and (20)) would not be feasible for moderately complex structures.
TMI-1 UFSAR APPENDIX 5C 5C-16 REV. 18, APRIL 2006
APPENDIX B Earthquake Analysis
TMI-1 UFSAR APPENDIX 5C 5C-17 REV. 18, APRIL 2006 APPENDIX B
EARTHQUAKE ANALYSIS
The computation of seismic stresses was carried out on the basis of the fundamental mode of the
containment structure associated with maximum response. The peak of the response curve (0.31g)
for 2 percent critical damping and 0.12g peak ground acceleration was used to determine:
(1) The stress-resultant Nf at the center of the opening (in the shell without the opening) for the horizontal component of earthquake motion oriented in the direction normal to
the openings.
(2) The in-plane shear stress-resultant NfP at the center of the opening (in the shell without the opening), for the horizontal component of earthquake motion oriented at
900 with the direction normal to the opening.
(3) The stress-resultant Nf at the center of the opening (in the shell without the opening) for the vertical component of motion associated with 0.12g peak ground acceleration.
The influence of the opening on the above seismic loads was evaluated as follows:
- a. The stress-resultant and stress-couple distributions and, therefore, the stress-concentration factors corresponding to (1) and (3) were conservatively
computed on the basis of the finite-element results for dead load.
- b. The stress concentration factors corresponding to (2) were determined on the basis of Lekerkkerker's solution (Reference 11) (See Figure 3) for a shell with a hole
subjected to torsion, i.e., to a pure membrane shear NfP at the location of the
opening, (in the shell without the opening). Note that the stress concentration factors
are slightly larger than those corresponding to the plate solution. In computing
stresses at elements away from the edge of the opening, however, the stress concentration factor was assumed to decrease as in the plane solution.
In determining the values shown in Table (6) and (7) the absolute value of the contribution of the
vertical component of motion (3) was added to the absolute value of the contributions of the two
horizontal components [(1) and (2)].