ML20008D774
| ML20008D774 | |
| Person / Time | |
|---|---|
| Site: | Midland |
| Issue date: | 01/13/1969 |
| From: | CONSUMERS ENERGY CO. (FORMERLY CONSUMERS POWER CO.) |
| To: | |
| References | |
| NUDOCS 8007300670 | |
| Download: ML20008D774 (4) | |
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(d AFFEIDIX 5F REACTOR bUILDII;G STRUCTURAL ANALYSIS STEESS A"ALYSIS The finite element technique is a general method of structural analysis in which the continuous structure is replaced by a system of elecents (merbers) snr.ected at a finite nuber of nodal points (joints). Conventicnal analyses of frames and trusses can be considere1 to be examples of the finite element methed. In the application of the =ethod to an axisy= etic solid (eg, a concrete reactor building), the continuous structure is replaced by a syste=
of rings of triangular cross sections which are i'nterconnected along circum-ferential joints. Based on energy principles, force equilibriu= equations are fomed in which the radial and axial displace =ents at the circu=ferential joints are the unknowns of the system. A solution of this set of equations is inherent in the solution to the finite ele =ent system.
There are many advantages to the finite element =ethod, when ccepared to other numerical approaches. The method is cc pletely general with respect to gecretry and =aterial properties.
Cc= plex bcdies cccposed of many dif-ferent r'terials are easily represented; therefore, in the analysis of the reactor building, ecccrete, and foundation =aterial can be realistically considered. Also, arbitrary ther=al, mechanical and gravity leading can be analyzed.
s The finite element method assu=es a ec=patible or semicc=patible displace-ment field for a typical cross-sectional area (element - cross section of ring). The nu=ber of degrees of freedc= deter =ines the number of unkncvn generalized displace =ents of each element - nedal point and consequently the nu=ber of generalized forces through the stiffness equation.
The accuracy of the results depends on:
The shape of the element (equilateral triangles are the = cst advisable, whereas obtuse triangles are unable to give correct soluticas).
The assu=ed displacement field, (linear, bilinear or higher degree).
The nu=ber of nodal points per element.
The size of the elements.
The total nu=ber of ele =ents and nedal points (round off errors of ec=puter in the solution of simultanecus linear equations).
FIhm EIFE iT CC.AM4 PROGRAM The initial develop =ent of the ec=puter prograa used in the analysis of the (m) reactor building was conducted at the University of California at Berkeley v
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in 1962 under a National Science Foundation Grant. Since ths; time, the progra= has been further modified and refined by Dr. Edvard 1. Wilson. The validity of the specific progra= used in the reactor building analysis has been established by the analysis of axis /==etric solids with kncun exact linear solutions. The stresi field given by the finite ele =ent =ethod satisfies internal equilibriu= and the boundary conditions involving stresses with a de6ree of accuracy directly proportional to the stiffness = atrix of s
each ele =ent.
In turn, this stiffness = atrix cnly considers a finite nu=ber of degrees of freede= for the displace =ent and consequently a finite nu=ber of generalized forces statically equivalent - frc= a virtual verk point of view - to the existing equilibriu= stress field.
j Another aspect of the finite ele =ent =ethod - which is very i=portant and often overicoked is the displacement ec=patibility between each ele =ent.
This question becomes essential vnen the size of the ele =ents is relatively large. It should be noted that the displace =ent ec=patibility along the j
ele =ent boundaries can be satisfied partially or cc=pletely depending on the assu=ed displace =ent field.
CCMPARISON WITH ICIOWN SCIUfIONS An exact analysis of the reacter building under consideration is i=possible by classic =ethods. A preliminary approxi= ate analysis of the structure is conducted, based on classical shell theory. In addition to the difficulty in representing the liner, reinforcing rings and fcundation caterial, shell theory neglects thickness and shear defc:=ations. Since the finite ele =ent approach includes thickness and shear defc:=ation, an exact ec=parison with shell theory cannot be expected. However, cross-section forces obtained l
frc= the finite ele =ent =ethod at sections not near rings or the foundation l
should agree with the results based on shell theory.
An analysis of the reactor building is done according to general shell theory for hc=ogeneous surfaces of revolution. The matrix of influence coefficients i
(generalined displace =ents due to generall:ed unit forces) are solved for the condition of equal deflection and rotations. Similar analysis =ethods l
are used for the intersection of the base slab and the cylinder vall. The base slab is analyzed as an elastic plate on a rigid foundation.
The results thus cbtained are expected to be within five percent of those obtained by a core rigorous finite ele =ent progra= for ring 61rder, dc=e and cylinder vall of a similar reactor building.
For the base slab analaysis, using the finite ele =ent progra=, the matrix is extended into the ground to account for the variations in soil character-1stics.
ANCHOSAGE ZONE STRESSES
-For dete =ining the anchorage zone stresses as well as the stresses in the nearby region of anchorage caused by hoop prestressing forces, the folleving procedures are e= ployed:
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his method is based both cn experimental photeelastic results as well as On equilibriu Ocasiderations of h::cgeneous and ecntinucus media.
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is used in a plane strain approach. As the prcblem is three dimensional, plane strain analysis is a better apprcximati:n than plane stress analysis. E:vever, as the progra: is prepared for plane st ess analysis, the redulus of elasticity, I, vill te 2.5 in Order to have changed to,
and Poissen's ratic, v, to planestrain_]effect.
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with the =axi=um tensile strain at crackin6 being increased.
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%J The considered buttress anchor cnes are subjected as such to additional vertical stresses, leading to a pseudo biaxial state, the se:Ond directic:
being radial through the thickness.
Fcr these reascas, special anchorage :One reinforcing is used in additica to that required by the leading cases.
Such special reinforcing is based on the following considerations:
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Pall scale lead tests of the anchcrage en the same concrete =ix used in the structure and review cf prior uses of the anchcrage.
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The post-tensicaing supplier's rece=nend.ations of anchorage reinforcing require =ents.
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