Convergence Criteria for Numerical Methods

70 Pages·1.554 MB·English

by  Levine, Bernard

THE SENEHAL INSTABILITY AND DESIGN OF CIRCULAR
CYLDBRICAL SHELLS REINFORCED WITH RING FRAMES
UNDER HYDROSTATIC PRESSURE
DISSERTATION
Submitted in partial fulfillment
of the requirements for the
degree of
Doctor of Philosophy
at the
Polytechnic Institute of Brooklyn
by
Bernard Levine
September 1951
Approved;
Head ofjDepartment
ProQuest Number: 27591394
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Approved by the Guidance Committee;
Major: Applied Mechanics
floJ, Hoff (Chairman)
Professor and Head of Department of
Aeronautical Engineering and
Applied Mechanics
Minor; Mathematics
'4P, JSussell
Professor of Mathematics
Minors Aeronautics
PoA. Libby
Professor of Ajéronautics
TO D.G.L
BIOGRAPHICAL SKETCH
The author was bom in Brooklyn on October 7> 1923* He received
his primary and secondary educations at Public School 122 and Boys
High School, He then attended the Wasington Square College of Arts
and Sciences of New York University where his major field of study
was in Physics* He received a Bachelor of Arts Degree from the
University in June 1945* He entered the Polytechnic Institute of
Brooklyn the same year as a research fellow. In June 1948 he earned
the degree of Master of Science (in Applied Mechanics). The title
of his thesis was "Convergence Criteria for Numerical Methods". He
continued his graduate study in order to qualify for the Doctorate
in Applied Mechanics while employed at the Polytechnic. The research
described in the present paper was initiated in the Spring of 1950
and was part of a project sponsored by the Office of Naval Research*
The writer is co-author of the following publications;
Hoff, NoJ., Mandel, M.W., and Levine,B.; "Torsional Instability of
a Box Beam Subjected to Pure Bending", April 1948, Polytechnic
Institute, to be published by.the NyAoCoA*
Salerno, V.L*, Liebowits, H*, and Levine, B*: "Numerical Analysis of
Sweptback Wings", 1949, to be published* , «
Salemo, V.L. and Levine, B.i "Buckling of Circular Cylindrical Shells
with Evenly Spaced^Equal Strength Circular Ring Frames, Part I,"
Polytechnic Institute, PIBAL Report No* 167, April 1950*
Salerno, V*L. and Levine, B. ; "Buckling of Circular Cylindrical Shells
with Evenly Spaced, Equal Strength Circular Ring Frames, Part II,"
Polytechnic Institute, PIBAL Report No* 169, June 1950*
Salemo, V,L., Levine, B. and Pulos, J.G. ; "Charts for the Determination
of the Upper and Lower Limit of Hydrostatic buckling Pressures for
Reinforced Circular Cylindrical Shells," Polytechnic Institute,
PIBAL Report No, 177, November 1950.
Salerno, V.L,, and Levine B*; "The Determination of the Hydrostatic
Buckling Pressures for Circular Cylindrical Shells Reinforced with
Rings," Polytechnic Institute, PIBAL Report No. 182, Feb. 1951°
AGKNOWLEDGEMENT
The author would like to extend his gratitude to Dr. K.J. Hoff
for suggesting the thèsis topic and for his inspiring guidance through­
out the course of this research and the pursuance of the graduate pro­
gram* In addition, he would like to thank Dr. V.L* Salerno and the
PIBAL Staff for their cooperation.
Since the research reported on in this thesis represents one phase
of work on a contract with the Office of Naval Research the author expresses
his thanks for the monetary aid extended by this organization.
ABSTRACT
The Rayleigh-Timoehenko method has been applied to determine
critical pressures for the general instability of circular cylindrical
shells reinforced with ring frames, under hydrostatic pressure. Both
simple supports and edges:: which are clamped against rotation at
bulkheads are considered. In addition a direct design method is de­
veloped. This method considers the yielding and buckling of the unit
bay as well as the general instability of the complete structure* A
numerical example is given which indicates a saving of between 58$
and 72$ in the weight of the ring frames as compared to standard
design.
TABIE OF CONTENTS
Page
Symbols
Introduction 1
Summary of the Strain Energy Expressions 3
Assumed Buckled Shape - Simple Supports 6
Determination of the Critical Pressure 10
Assumed Buckled Shape - Clamped Edges ^ 17
Discussion of Approximations 21
Minimization with Respect to A, 22
Numerical Example I 23
A Direct Design Method 25
Shell Design 26
Ring Frame Design - Simple Supports 29
Ring Frame Design - Edges Clamped Against Rotation 33
Refinements of the Design 35
Numerical Example II 40
Comparison with Standard Procedure 46
Conclusion 48
Tables 49
Appendix I 52
References 54
Figures 56
TABLE OF SYMBOLS
is the area of the ring frame section
bj hj t* are dimensions of the ring frame
is the St. Venant torsion constant
e is the eccentricity of the centroid of the ring frame section
E is the modulus of elasticity
G is the shear modulus
h is the thickness of the shell
I , I_ ] are moments of inertia,of ring section
o 2
k - hV12R
K - Eh(l- )
Ljj is the distance between bulkheads
L^ is the distance between ring frames
I-o- V »
m is the number of waves in the circumferential direction
* # * * #
Q, T, M^, Mg, T j Q , N are ring frame dimension parameters
P is the axial pressure in lb* per sq. in,
p is the radial pressure in lbs, per sq. in,
q is the number of half waves in the radial direction
R is the radius of the shell
®A' ’*6» ®C1* **C2» ®AC ®BC \ ^AC ®BC frame energy parameter*
R^, Rg, R^, R^ are trigonometric summations
S « e/R
u, V, w are displacements of the shell
Üq, Uy, Uy are strain energy expressions
V^, Vg are potential energy expressions
X, y, z are coordinates of the shell