-
1
OPTIMUM COLUMN STIFFENING
by
LUCIEN ANDRE SCHMIT
S.B., Massachusetts Institute of Technology
(1949)
Submitted in Partial Fulfillment of the
Requirements for the Degree of
MASTER OF SCIENCE
From the
Massachusetts Institute of Technology
(1950)
Signature Redacted
Signature of Author
Department of Civil and Sanitary Engineering
January 14, 1950
Signature Redacted
Signature of Professor
in Charge of Research................
Signature of Chairman of Department
Committee on Graduate Students .....
.*...
.090.
Signature Redacted
......
e..
ACKNOWLEDGEMENTS
The author wishes to express his gratitude to
Dr. C. H. Norris, whose patient, considerate and
lucid instruction made this thesis possible.
The
author would further like to take this opportunity
to thank Mr. Stanley Falconer and Mr. Harlow Farmer
for their helpful suggestions.
i
TABLE OF CONTENTS
Page
Introduction
B.
Purpose
C.
Scope
D.
Method
E.
Results and Discussion
F.
Conclusions
- - - - - -
-
1
- - - -
-
- - - - - - ---
A.
Preface
- - - - - -
-
INTRODUCTION
B.
Purpose
- - - - - -
C.
Scope
- - - - - - -
D.
Method - - - -----
INVESTIGATION
10
B.
Differential Equations of the Generalized System
11
C.
General Solution of the Characteristic Equation
15
D.
Optimum Curves for Circular and Square Tubes -
18
E.
Minimization Criteria
F.
Optimum Curves for a Structural Cross Section
27
G.
Results and Discussion v - - - - - - - - - -
36
H. Conclusions
- - -
-
-
-
"
- -
- -
w
-
-
-
Formulation of the Prol lem
-
- - - - - - - - -
A.
-
III.
A.
-
iI.
SYNOPSIS
-
I.
23
42
I
ii
Page
IV.
APPENDICES
A.
Notation
B.
Discussion of Dr. Dinnik's Paper --
C.
(1)
- - - - - - - - - -
44
------
46
Computations for the Plot which Represents
the General Solutions of the Characteristic
- - - - - - ----
Equation (13)
(2)
(3)
51
The Plot of rf-L
versus q for k values
or
ranging from 0.5 to 1.00 - - - - - - - - - Cross Plots of
Values of
k
4
versus
"--
q
64
for Constant
--------
65
cr
D.
(1)
Computations for the Optimum Curves for
Circular and Square Tubes
(2)
.. 82.
-
from Table II
- -
o
70
-
- - - - - - - - - - ---
--
83
-
Computations for Minimization Criteria for Cirocular and Square Tubes-
H.
-
-
Computations for Spot Check of Conclusions Drawn
- - .----- - - ---a 85
G.
-
Algebra of Method for Checking Validity of
Equation (26)
F.
-
Plot Showing Computed Minimum Points and
Average Optimum Curve Drawn
E.
-
--------
-
-
86
Computations for Minimization Criteria for
Structural Cross Section
BIBLIOGRAPHY
- - - ----------
--- ------
--
-
-
-
91- - W M - " - t.
88
91
6
iii
LIST OF FIGURES AND TABLES
Figure
Page
1.
Schematic Drawing of Generalized Stiffened System
2.
Schematic Drawing of Circular and Square Tubular
Cross Sections-
3.
Typical Plot of
&V
11
-
---------------
versus
qk
18
for Constant I-
24
cr
4.
Schematic Drawing of Structural Cross Section
5.
Plot Representing General Solution of the Char-
acteristic Equation (13) .6.
-
-- -- -- -- --
-
-
28
--- -- 64
Plot Showing Location of Computed Minimum Points, and
Average Optimum Curve Drawn
--
-
--------
-
82
Table
1.
Table of Minimization Criteria
G(k,B)
2.
Table of Minimization Criteria
G(kB) and g(kc,D)-.
-
-
-
-
-
26
34
1
I.
A.
SYNOPSIS
Introduction
The basic ideas for this investigation were stimulated by the
common limitations placed on allowable -stresses in Euler range col.umns of constant cross section.
If the first step in the design of
an Euler range column was to select a cross sectional area sufficient
to carry a desired load in direct stress, the second step would be to
prevent the column from buckling.
The buckling can be prevented by
the addition of material which increases the moment of inertia of
the column
cross section at various points.
called stiffening.
Such added material is
There is an optimum manner in which to distribute
the stiffening material for any particular column design.
A design
method which is more efficient in the use of material than current
constant cross section practice can be devised.
B.
Purpose
It is the purpose of this investigation to devise a readily ap..
plicable method for determining how to stiffen most efficiently a
column loaded above its initial buckling resistance, against buckling
failure.
C.
Scope
This investigation is limited to axially loaded, Euler range
columns, with pinned ends.
stant cross section.
The unstiffened initial column has a con-
The initial column and the stiffening are of
2
the same material, and have cross sections with two axes of symmetry.
The stiffening used has a constant cross section itself, and is ap*.
plied continuously over some fractional length of the column.
Fin-
ally, it is assumed that the stiffening material acts integrally with
the initial column as if there were no interface between them.
D.
Method
For a generalized stiffened column system within the scope of
this thesis, a differential equation solution is carried out, which
results in a characteristic equation.
A general solution of the
characteristic equation is developed algebraically and presented
graphically.
Curves of minimum increase in material solutions,
optimum stiffening curves, are developed for circular and square tubu.
lar columns, by a method of trial computation.
The idea of minimiza-
tion criteria is introduced, and optimim stiffening curves for a
structural cross section are related tothose already obtained for tubular columns,
E.
Results and Discussion
The graph of the general solution of the characteristic equation
of the simple stiffened column system defined has been developed.
An
average optimum stiffening curve is superimposed on the general solu-4
tion of the characteristic equation.
(See Figure 5, Appendix C (3).)
For columns which conform to the limitations, and are within the ranges
and of the types of cross section considered, the optimum stiffening
solution may readily be computed using Figure 5.
Although the location of the optimum stiffening curve depends on
the shape of the cross section in a strict theoretical sense, the results of this investigation show that the locations of the optimum
curves, for various cross sections, do not vary appreciably.
troduction of an average optimum curve is reasonable.
The in.
The average op.*
timum stiffening curve presented in Figure 5 may well be applicable to
stiffening problems involving far more complicated structural cross
sections with tvAo axes of symmetry.
F,
Conclusions
From the results of this investigation, it may be'concluded thatt
1.
A readily applicable method for determining the optimum stiffening
solution for a column which is within the limitations, and has a cross
section of the type, and within the ranges considered, has been dew
vised.
2.
Although savings of material are achieved by optimum stiffening
design, the cost of fabrication probably outweighs any economy of
material.
In designs where weight is a critical factor, optimum col-.
umn stiffening might well find application.
3.
The ideas successfully applied to the simplest type of column in
this investigation may be applicable to a large number of more com-
plicated column types.
II.
A.
INTRODUCTION
Preface
The basic ideas for this investigation were stimulated by the
common limitations placed on allowable stresses in Euler range columns of constant cross section.
The reason for such reductions in
allowable stresses is that buckling resistance governs the capacity
of a long column, rather than resistance to yield in direct stress.
As a result of being limited in strength by buckling resistance,
none of the material in the column is used efficiently, inasmuch as
it is not stressed to its allowable limit.
If the order in which the two strength requirements are considered is reversed, a design method which is more efficient in the
use of material can be devised.
Then the first step in the design
of a long column would be to select a cross sectional area sufficient
to carry a desired load, stressing the material to the allowable
limit.
The second step would be to prevent the column from buckling.
Assuming there is no reduction in the unsupported length of the col..
umn, buckling can be prevented only by the addition of material to
the column.
This addition of material which increases the moment of
inertia of the column's cross section at various points is called
stiffening.
B.
Purpose
It is the purpose of this investigation to devise a readily appe
plicable method for determining how to stiffen most efficiently a
5
column loaded above its initial buckling resistance against buckling
This problem has two major divisions.
failure.
1.
What are the possible ways of increasing the buckling re-
sistance of a column by stiffening?
2.
Which of the possible solutions shows the greatest economy
of material?
C.
Scope
Although the concept of column stiffening is applicable to all
columns, a thorough investigation of only one very common, but par
ticular type of column, has been carried out.
The type of column here considered in detail can be described
as followst
1.
Load:
The column is loaded axially without eccentricity and is
assumed to be ideal.
2.
Ranges
(Perfectly straight, homogeneous, and elastic.)
The column is in the Euler column range; therefore the
material is perfectly elastic and no stresses occur beyond the propor"
tional limit of the material considered.
If the material is steel$
then this investigation is limited to members having an r
ratio
equal to or exceeding 100, where L is the length of the column, and
r
the radius of gyration of the unstiffened column's cross section.
More generally, this investigation is limited to columns having an
ratio equal to or exceeding TT 4 fL
, where E is the modulus
?.L.
of elasticity, and fP.L. is the proportional limit stress intensity
of the material used,
6
3.
End Conditionst
The column is pinned at both ends, making it
symmetrical about its longitudinal midpoint.
A pinned end allows
free rotation but no lateral translation at the end of the member,
4.
Type of Stiffening:
The stiffening used has a constant cross
section itself and is applied continuously over some fractional
length of the column.
It is physically clear that the most effic".
ient distribution of the stiffening, for symmetrical end conditions*
will be symmetrical about the column's longitudinal midpoint.
There..
fore, only symmetrical distributions of the stiffening material are
considered.
5,
Modulus of Elasticityt
It is assumed that the modulus of elas..
ticity of the unstiffened column and that of the stiffening material
are the same9
6.
Cross Sectiont
The column and the stiffening are assumed to have
cross sections with two axes of symmetry.
7.
Integral Action:
It is assumed that the stiffening material acts
integrally with the initial column,
tween them.
as if there were no interface be.
The design of a fastening system that will make this
assumption a reality is a considerable problem in itself*
D.
Method
The method used in this investigation will now be outlined.
An
expression relating the buckling resistance of a stiffened column to
the amount of stiffening material, and the given properties of the
unstiffened column, is derived from the fundamental differential
7
equationt
El dy
1
M
dx
This equation of the beam theory makes the simplifying assump..
tion that
d22
[ + (Z)233/2
dx
where P is the radius of curvature of the distorted structure. This
assumption is sound for small deflections because ( )2 is small com.
pared to unity. However, when equation (1) is used
in the solution
of buckling problems, it
leads to a result which implies that buckling
failure is a discontinuous phenomenon wherein the lateral deflection
instantaneously becomes infinite. The buckling failure of a structure
is a continuous physical phenomenon, and a solution based on the exact
expression for the radius of curvature would verify this mathematically.
See Chapter I, Friedrich Bleich, The Buckling Strength of Metal Struc..
tures.
The relation between the amount of stiffening, the given properties
of the unstiffened column, and the buckling resistance of the
stiffened column, is a transcendental characteristic equation.
It can be reasoned physically that the buckling resistance of a
stiffened column is equal to some constant, determined by the properties of the unstiffened column, multiplied by a function of the
length and bending stiffness (EI) of the stiffening.
Physically,
then, there are three variables involved, namely the buckling re"
sistance of the stiffened column, the length of the stiffening, and
the bending stiffness of the stiffening; any two of which may be in..
dependent variables.
Certain convenient changes of variable are made
8
in the derivation, but the resulting transcendental characteristic
equation contains three variables which are related to the physical
variables discussed above.
A graph representing a general solution
of the characteristic equation in terms of three dimensionless var.
iables, which are clearly related to the physical variables, is de.
veloped.
(See Figure 5.)
Given a column with an Euler buckling resistance Pl, it is deo*
sired to increase its buckling resistance to Por by stiffening.
There are
n
solutions to this problem, given by any point on the
horizontal line dramat
fT7
'
is defined as
.
(See Figure 5.)
Note that
k
or
where
1
equals the moment of inertia of
12
the unstiffened cross section, and 12 equals the total moment of
inertia of the stiffened cross section.
Let
q
equal the ratio of
the stiffened length to the total length of the column.
value of
of
q
k
to the right of
For each
k n 0.5, there is a corresponding value
which is a solution giving the required increase in buckling
resistance.
.
Values of k to the left of k a 0.5 represent stiffening by
values of 12 > 411 and are Lnot considered. The initial column
must have an extremely high yR- ratio, to have an optimum stiffenow
ing solution for which k<0.5,
that is 12
In order to determine which of the
each value of
jL
n
available solutions for
is a minimum volume solution consider an exoo
or
pression for
AV,
the volume of the stiffening material.
Let A 1
9
equal the area of the unstiffened cross section, and A2 equal the
total area of stiffened cross section.
A
.
AV W (A2 "0 A)1 n ) ( A 1) qIly
A2
The relationship between the area ratio -
2
Il
= k
12
ertia ratio
,
and the moment of in"
A
1
of the cross section.
depends on the shape
However, for various cross sections having two axes of symmetry
A2
can be expressed in terms of k and certain constants related
A1
A2
can
to the shape of the unstiffened cross section, Therefore A
-
be written as
h(Kk), where
h
means a function
h
of k and Km
is a certain constant depending only on the shape of the in.
and K
m
A2
itial unstiffened cross section.
AV
expression for
Substitute h(Kmpk) for
in the
above.
A2
AV
q
[h(Km k)
Having expressed
JV.
variables
the
w 1) qLA 1 - [h(Km k) - 1] qAjL
is a constant,
Since IA
of
(2
and
k
and
k
-
1) q
AV in terms of the basic dimensionless
this expression and compute
AV.
n
ji-i
we substitute
or
available solutions in
q, for any fixed value of
values for each of the
a minimum value to
can be taken as a measure
The value of
k
and
AV is called the optimum point.
q that gives
If such optimum
and a curve
points are found for a number of fixed values of
or
is passed through them, it will be a curve of minimum
AV solutions.
A curve of such minimum volume solutions for any particular initial
cross section is called an optimum stiffening curve.
(See Figure 5.)
10
III.
INVESTIGATION
Formulation of the Problem
A.
Consider an Euler column design for a direct load on a column
of length
L, pinned at both ends, and with a cross section having
two axes of symmetry, cross sectional area A1 , moment of inertia Il,
Then Euler's formula gives the buckling resistance of the column as
PTI
2EI
2.
L
criteria.
The capacity of the column is limited by the buckling
The direct stress intensity
P
f 1 =A-
able stress intensity
f
m=
a
SF.
is below the allow.-
1
f Y*
, where f
is the yield point
Y?
stress intensity of the material used, and S.F. is the safety factor
with respect to direct stress intensity.
if
load by a factor (
Suppose we increase the
if
) so that P2
a
1
, then the stress intensity
will be up to the allowable limit fa , and we will be making full
use of the material.
load P2
1
,
Of course, the column will buckle under the
unless it is stiffened.
It would be reasonable
to design the stiffened column to carry an ultimate load
Por n S.F. P , where S.F. is the safety factor with respect to
Vuckling resistance.
This is only one rational method of arriv.
ing at a value for the ultimate buckling resistance Por to which
the column is to be stiffened.
Regardless of how P
cr
is prescribed,
or
*
the problem is how to prevent buckling by stiffening under a load
11
B.
Differential Equations of the Generalized System
L
II
Is
cL
Figure
1.
The general fundamental differential equation is
2
EI
.0
Yd-x " M
0 0
0
0.
0
0
0
0
(1)
dx2
Then over the unstiffened portions (region 1) the following equation
is valid:
2
EI
d
dx Yl
PrYl
0
0
(2)
0 0 0 0 * 0 0 *0
Over the stiffened portion (region 2) the following equation is
valid
2
d y
EI2
dx
cr
12
P
Let
Consider the general solution of equation (2).
Then
k
or
equation (2) can be written as
2
d y1
dx
+ k1
2
y
O-0
1
The general solution of equation (2) is
ofcos k x +
Y
1
1
at2 sin k1x
Applying the boundary condition x
(4)
.........
0 .,y1 - 0
to equation (4)
ofn 0
0 oic 1 +0
1
and
Y nsin kx0. 2. . . . . . . . . *1. . (5)
P
Let
Consider the general solution of equation (3).
.
k
-
El 2
Then
equation (3) can be written as
2
d y2
.
2
-,-+ k 2 y7
2 O
dx
The general solution of equation (3) is
y-
cos k
x +
c sin k 2x
.
.
Applying the second independent boundary condition
.
.
X
.
.
.
.
(6)
2
Sdy2
to equation (6)
0
I
13
dy2dy
M31
-c
0
sn
k2 sink2 x
2
x
2
k 2 TL
sin k
k
2
cos k
L
22
L
cos kZ
tnkL
mt
tank2
x + o
tan
k2 L
2
sin k
* *
*
x
L
Applying the third independent boundary condition
yl ' y2
*
(7)
to equations (5) and (7),
c osik
sin k, L (1 - q) = o
cos k 2
(1 -q)
+ of tan
kL
k2
L
L
sin k2 7 (1
. (8)
Applying the fourth independent boundary condition
Sdy
2
-
x
-
(1
q)
,
$ (1 q),
4
cook
-ct k sin k2
3 2
2 2 + c4 k 2 co
c,
0
and
+c' k
.
~
to equations (5) and (7),
kof cos kL (l.q) - - c
k2 sin k
(-q)
k132
2 -7
1 2
+Ot k
+
2
k2L
008k 2ktan*2 * *(9)
Divide equation (8) by equation (9) and multiply both sides of the
resulting expression
by k2
k2
1
L
1 + tan -,- tan k 2 T (1
L
w tan k2 E (1 - q) + tan
q)
k2 L
. ..
(10)
q)
1
14
Employing the trigonometric identity
)
tan (a
tanA
tanC,
1 + tan(C
tang
equation (10) can be written
k2tan k
1
L (1 - q)tan (
..- k2
or, simplifying,
tan k
Let k
k
--2
(l - q) tan
k2 L
m
s.-
.
.
.
0
0
0
0
*.
(11)
cr
a
0
112
Then, replacing k1 by
Mr,
and k 2 by
kPJCr
,
equation (11)
becomes
(1 ~ q) tan k
k tan
if we define a parameter U, such that
U
q) (1-q) T0
then equation (12) can be expressed as
g q
1..
. .. (12)
-
tan U tan
-
-.
_
U
1.q
_
___
_15
.(13)
k
where
(14)
Srq)
or solving for Pcr
2U
1
or
EI1
2
q
L2
Equation (13) is the characteristic equation of the generalized stiffened column system considered.
values for k,q, and P
or
Any set of real positive
which satisfy equation (13) constitutes a
6
solution to a specific stiffening problem.
This derivation is out.
lined by Dr. A. N. Dinnik in his paper on columns of variable cross
section.
C.
(See Appendix B.)
General Solution of the Characteristic Equation.
The characteristic equation contains three variables, any two
of which are independent, the remaining variable being dependent.
If
q
U
and
are considered to be the independent variables, and
k
the dependent variable, the problem of transcendental solutions
is avoided.
If
algebraically,
U
are assigned values,
and k
q
may be computed
It will be both convenient and general to plot the
general solution of the characteristic equation (13) in terms of the
k
,
-
12
q,
and
,
dimensionless variables
or
----------
16
if 12 is limited to values less than 41I, then the range of
k values is
0.5
k a
k
M 1.00
-
k range
0.5 -+*
-
1.00
0.500, 0.525, 0.550,
k,
For each of the ffollowing values of
0.575, 0.600, 0.650, 0.700, 0.750, 0.800, 0.850, 0.900, 0.950; U
are assigned,
values from zero to -
\1 or
q, and
and
are
Com.*
puted.
Use equation (13) to compute q, given k and U
00
*
*
0
0*
*
(13)
*a
0
.
U
q
Uta
ta
Use equation (15) and the expression P
to determine the
a
can be computed, given
following equation (16), from which
or
k and U and q,
2 EI1
ZU
or
1 0 q
P1
2
ST(1
r2U
or
L
L
EI
2
2U
T-(1 - q)
---
-d)
0
0 0
0
0
0
0 .0 0
0
0
0
(16)
17
(See Appendix C (1).)
Sample Computation:
U a 0.2000
k a 0.950
Using equation (13)
U a
tan U tan
k
0.2027 tan 1 **q 0.1900 a 1.0526
0.1900. = 5.1928
tan
5.1928 a 1.38055
0.1900 n tan
T
-
7.2660
7.2660
n
1 + 7.2660
q = 0.8790
T(1 ~q)
2U
cr
\
V
(1 ~q)
2U
-if
(1 * 0.8790)
W~
or
r(i
,
Using equation (16)
2 x
a
0.2000
3.14lg
0.9505
0.1210
*1273
0.9505
18
D.
Optimum Curves for Circular and Square Tubes
For an unstiffened column with a buckling resistance P, it is
desired to increase its strength to Por
There are
n
stiffening the column to a buckling resistance P .
cr
ways of
These
n
solu-
tions are given by any point on a horizontal line drawn at
or
In order to determine which of these
(See Figure 5, Appendix C (2).)
n
solutions is a minimum volume solution, an expression for
terms of
k
and
q
AV
in
must be developedt
A
V
Since IA
-
(A2
A
211 ) Lq -
(). 1
)qL
. .. *
.
1
is a constant, it remains only to express
(17)
as a function
AW
of
k
or
q.
The relationship between
V-
and k
depends
2
on the cross section shape,
A
The derivation of the expression for
-
2
in terms of
A1
now be carried out.
-4
-
C11
C(I
I
'I
CIO
K
K
Figure
2.
k
will
19
For circular tubes,
For square tubest
I,
I
2
I
(d24 P d
T
I
A1 = d 2
0~L~
14
A2
to k
To relate
let
d
t q -
let
B
F12
40
1
d
.d
d
and note that
0
t
B
2(B + 1)
Then for circular and square tubes
4
(d
( 4 . do)
4
4
(d 2 m do
I1
-
k
2
4
4
2
d 4
1
d
2
B
-
d
or
0
Therefore*
I
0
0
*
But
(d 4 # d 4)
1
a 1
d
(d 24
2
12
2
A2 nd2 2 *d
d
o
(d2
A2
-
1 12
2
o d
)
(d
)
ow d
0
2
.. d 4
)
A,
64 (d1
d
0
-4-
1+ B
0
0
20
4
4
d
d1
(1 + B)
k2
4
d (1
2 + B) 4
1
(1+ B)
d2
2
d2
22
1
(1 + B)
4
d
d
1-
(1 + B)
1
(
(1 + B)
k2
.
k (
1
(1 + B)
k (1 + B)
4
(1 + B)4 " 1 + k2
k- (1 + B)4
(1 + B)4 - 1 + k
d
k(1 +B) 4
But
2
A2
17
A1
d0
(d 22
a
d 2
1
A2
A
(
(1+ B)
1
A2 _ 1/k
2
+ k
4
(1+B)
d_2
k (1 + B)__
d 2)
(d
42
d1
(1 + B)
d
(1 + B)
kl
+ B)4
k(1 + B)
2
(1 + B) 21
(1 + B)4
1 + k2
1
0
A
B
+ 2B
0
0
*
0
* (18)
21
A
Substitute for
from equation (18) into equation (17):
AV
k
.
B
q LA1 . . (19)
+ 2B
Equation (19) is an expression for
AV in terms of the variables
k and q.
Recall that for each particular value of
we had a set
or
of
as
n
solutions in terms of
d
d0
q
and
k.
The constant
B
defined
is a convenient constant which depends on the
0
t
of the unstiffened column,
lower limit of
t
1
If an upper limit of
-
ratio
d1
1/3, and a
is adopted for consideration, the range of
practical tubular columns is completely covered.
These limits were adopted after a study of available tubing as
indicated by various handbooks.
See p. 15, Handbook of Welded Steel Tubing, Formed Steel Tube
Institute, 1621 Euclid Avenue, Cleveland, Ohio. January 15, 1941.
See Section I - 16, Summerill Aircraft Tubing Data, Summerill
Tubing Company, Bridgeport, Conn. March 8, 1943.
Localized buckling failure of the unstiffened portion of the
tube is not critical. Since this work is limited to the Euler range
the stress intensity in the unstiffened portion of the column will
never exceed the proportional limit.
See p. 440, Theory of Elastic Stability, Timoshenko.
2Et
cr
for steel
f
30,000
d
2
E = 30 x 106 -#/in2/r
22
t
30 x 103 a 2 x 30 x 106
3(1 - 1/9)
d
t
0.8165 x 10'
r
-
-
100
t
(
Since such thin tubes
1
)
10
are not practical, it is
clear that localized buckling of the unstiffened portion of the
tube is not critical for the columns considered.
L
For
,
B
-
2
and the expression for
AV (equation
(19)) becomes
k V80 + kT
AV
1
-
81
.J q LA1
.
. .
. .
(20)
- 0.975, 0.950, 0.925, 0.900, 0.875, 0.850, 0.825, 0.800,
For
or
0.775, 0.750, 0.725, 0,700, 0.675, 0.650, 0.625, 0.600, 0.575, 0.550;
a cross plot of
k
is drawn.
q
versus
(Appendix C (3).)
The ac-
curacy of these cross plots depends on the number of constant k
curves, and is the chief source of computational error in this work.
Then for each value of
the k
,
q
solution which makes
&V
or
a minimum is found by trial, using the cross plot and equation (20).
(See Appendix D (1).)
These minimum volume solutions are plotted on the graph represent"
ing the general solution of the characteristic equation.
A smooth
curve joining them is the optimum stiffening curve for circular and
t
square tubular columns with a
For 7- a
(equation (19))
17
,
becomes
B
11
1
and the expression for
AV
23
&y
. 0406 + k 21
0.0201
[/k
1
- 1]
q LA
The procedure is identical with that outlined for
that equation (21) must be used to compute
AV.
..
t
1
- 0 d1
(21)
.
except
(See Appendix D (1).)
The resulting minimum volume solutions are plotted on the graph representing the general solution of the characteristic equation.
A
smooth curve joining them is the optimum stiffening curve for cir-
t
1
t
.
cular and square columns with a
Due to the lack of accuracy in the cross plots of
k
versus
q
for fixed 4J!
values, the points representing minimum volume solu.
or
P
tions do not lie in the smooth curve. For each
value, the
or
curve of AV versus k,q solutions is quite flat, near the minimum
point fcr
t
AV.
The minimum points making up the optimum curve for
1
Salmost
the same as those for
1
t
1
-
It has there*
1
fore been decided to pass one. average curve through the two sets of
minimum points, which will be the optimum stiffening curve for cir.-
t1
cular and square tubular columns with a
between 1/3 and
0
(See Figure 6, Appendix D (2).)
E.
Minimization Criteria.
For the square and circular tubular columns considered, the
If the expression for
.
l /k
t
a-
1
and
t
1
-
&V, equation (19), is consideredj
(1+ B) 4
B
+ 2B
1 + k2
1
]LA
.
optimum curve is essentially the same for
24
recalling that for
t
1
-d
1
31
B = 2,
t
I
d
and for
B
2
1
100
it might be suspected from the form of equation (19) that the kq
solution which makes
AV a minimum is almost independent of B or
t
d1
It can be concluded from the computational work that
has only one maxima-minim& solution between
.
at each line of constant
q
-
0 and
A typical curve of
k,q solutions at any line of constant
AV
q = 1.00,
&V versus
is shown in Figure 3.
or
ASYMPTOTIC
cnV
-1
0
1.00
% ASSOCIATED
Figure
WITH
A
3.
Since there is only one real maxima..minima solution, d
be the criteria for this one solution.
-
o must
It must be established that
25
0
-
is the criteria for the one real minimum solution by the
reasoning above, because
q n f(k) for each line of constant
cr
is not known, except graphically.
If f(k) a q could be expressed
explicitly, then the conventional second derivative test could be
used to determine which of the solutions of
d
dic
0
was the real
minimum point, and also to show that this was the only real minimum
point.
Since
dk
- 0 is the unique minimization criteria, the exact
form of this criteria will now be determined.
with
q
replaced by
V
f(k).
1/k
(f + B) 4 - 1 + k2
B
[1/k V(1 + B)4 -1
AV -
Consider equation (19)
o 1
+ 2B
+ k2
1 -
J
B2 + 2B)1
f(k) LI
2
iaU
B +2B
dAV
(+B)
4
1 + k2 1/2+[(+B )4
Ik
+
Let
dAV
0
+ k2 ]/21
+2B
1)B
1/k [(l+B)4 " 1 + k2]1/200 1 - (B
and multiply both sides by
f (k)
) B + 2B
k 2 [(+B)4w 1 + k2
1 t(l+B) 4
0
+
1+ kc2 1 + k 23
f k
+
Then
(ic((l+B)4..+kcJ*(B+l)2kc2(l+B)4.4+i2l
/3
B Z +2
26
ivhiph can be reduced to
f(k) = G(kB) fr(k)
. . . . .
.0
0 0 0
(22)
where
G(kB)
-
k(+B) 4
1
+k2-(B+)
2 22 [(4+B)4-+k2 1/2)
1
(1+B)
. . . .(22a)
Equation (22) says that at any line of constant
where
or
q a f(k) is given graphically by the cross plot, the value of k such
that P(k)
f(k)
G(kB) is satisfied, is the minimum volume solution.
Each value of
k
from 0.5 to 1.00 is a minimum volume solution at
the appropriate value of
.
If for
k n 0.5, 0.7, 0.9, 1.00,
or
G(kB) is computed with B = 2 and then with B =
, using equation
(22a), the results can be tabulated as follows.
(See Appendix G.)
Table of Minimization Criteria G(kB)
1
&G(kB)
100
G(kB)ave
1.00
0
0.90
0.08995
0*08573
4.8
0.70
0,20972
0,18022
15.1
0.50
0.24961
0.19267
25.75%
%
%
0
Table 1.
A rough check on the G(k,B) values computed can be made. For a
value of k, say k
0.7, from the optimum stiffening curveppit can
be seen that k a 0.7 is a minimum volume solution between
n 0.725
cr
VL
27
and 0.750.
Draw tangents to the cross plots for
0.725 , 0.750
;
or
at the q value associated with k = 0.7 on the optimum curve, namely
q = 0.54. Then
q Ak
&
q
1''"kJdq
G(kB) a f
dk
G(kB) = q
k
0.59 x 0.20
.020
The fact that this rough graphic check falls between the two G(kB)
values computed indicates that everything fits together.
1 that is, as k
Table 1 shows that for lower values of
,
or
decreases, the minimization criteria for B a 2 and B
increasingly different.
-
become
However, for the f(k) functions involved3
a difference of 25.75% in the minimization criteria G(kB) doesn't
change the location of the optimum stiffening curve on the graph rep.
resenting a general solution of the characteristic equation appreci.-
ably.
F.
(See Figure 6, Appendix D (2).)
Optimum Curves for a Structural Cross Section.
The general solution of the characteristic equation is in-
dependent of the shape of the cross section.
plots of
k
versus
q
Therefore, the cross
for various constant values of
\f-i1
are
or
still valid. Then for a structural cross section using equation
A
(17), it is only necessary to express
as a function of k.
-
I
28
NOTATION.
t
Kb
t
Figure
4.
The cross section that is to be considered now is made up of
four angles arranged at the corners of a square and stiffened over
a leaph Lq by four plates.
Over the unstiffened portion of the
column, the four angles would be held together by lacing bars. With
A
the notation indicated in Figure 4, an expression for
in terms
of k will be dorived,
g(d --2b) t
1l4
t (d"2)
I
but t
and do
~(d
dc
d1 -.
2t
4
-d, )-"2 (d -2b)t(
b n tD
n cDd
(1
(l - 2c)
c)d d
d 1 +d
2
1
P) 2
3
1
3
29
(1 -
24
-2D
2(1*20D)
(1-2c)
2
3
c
1 + (1-2c) 1 2
c
+
Tim
2
3)
4
El s (1-2e)4],m6c(l-2eD)(1-c)2-2(1--2cD)c'3
'a
1
"I 2c (1 - 2oD)3]
d14
11 - Gd1 4
*...
.0
.......
.
.
(23)
where
G-
f[lo(1-2c)4]-6c(1-2cD)(1_c)2
-2(l.2cD)c3. 2c(1..QcD)13
. . . . .
I
is computed as IS + I
(d2 4 - d
4
S is the moment
,where
)
)23a)
of inertia of the cross hatched corners in Figure 4.
These corners
can not actually be provided if the stiffening plates are to be
welded to the angles.
assumed to be zero.
However, in this derivation, the term G is
This approximation is on the unsafe side, and
will be considered to trespass slightly on the safety factor.
1
12 = 1l + A I = Gd1 4 + Tr
0 assumed approximation.
I2 tG +
212
L
(
d1
4
-- d 4)u
Therefore
)4 0 ..
12
I d
1
. . *0
From the definition of k, and equations (23) and (24)
,
6-
4
*
(24)
--i
30
k
1
2
G
.-
[G + 1
A1 w
t1 - (1
A,
4c
A2
A
-
d
20)
2
[22D - 11 d
u1J+1
. . . . . (25)
- 2b)
_ 4c(1 - 2cD)] d1I
2
2
&A
[8Dc
A2
d
A2
A
d1 2 [8Dc
1
d
2
2
-
d 2
-. 4c
)
+ (d2 2
2
12G
4t (d
sd 2
A
A2 = A
3 + 1
k
d2 - d
2y 1 k 1
+
G
t41
12G
d1
1
TE
4
d
)
d2
)4
-
12
1
T2-
2
- 4e2
2
11] + d2
1] + d
4c (2D-1)
Using equation (25) to eliminate d2 from the expression for
A2,
A1
Then
-8Dc2
.s 4c2 .. 1
4c (2D 00
V12G 1C 2 . 1]
1)
+
2
+
1
1
31
A2J
+ H V12G[
1J + 1
2
. . . . .
(26)
.
.
.
(26a)
0
0
* 0 . (26b)
where
2
2
S8D2 (4c
..
1
2J "
402 (2D -d 1)
1 - i
.
.
and
H n1
0 .
0
0 0
4c (2D - 1)
This is the desired relation between
section under consideration.
A
A
for the cross
and k
In order to check this expression for
A2
d
in terms of k, consider the limiting case when b
,
that is,
the initial cross section of four angles is a closed square,
If for
this case we relate c and D to the constant B, using their respective
d1
definitions and the special condition that b
, (equation (26))
should reduce to equation (18).
This has been done, and since equation
(26) did reduce to equation (18), it constitutes a check of sorts on
the validity of equation (26).
An expression for
(See Appendix E.)
AV in terms of k and q can be developed as
follows:
A2
AV-
(
-
A11
1 ) q L A1
0 .
0
0
.
.
.
* (17)
Substitute from equation (26) into equation (17):
AV
n JJ + H 412G[ 1
1
+1
-1
q LA
(27)
k
where G, J and H are defined by equations (23a), (26a), (26b) re.-
32
spectively, and are dependent only on c and D.
In considering circular and square tulular columns, it was necessary to establish a range of -MM
t
limiting values
of vausofd
1
it
/
3 and
values, it
1
d1t
-g
-202
1tb
is now necessary to establish ranges for
will be recalled that
were adopted.
tb
Similarly,
c and
w D, the
1t
constants defining the shape of the initial
section,
The range of
4
rn
to
b
D n 20.
b
t
unstiffened columrft cross
D values considered here is from
b
D
These limiting values were arrived at by making
a survey of the angles available in the A.I.S.C. Manual of Steel ConO.
struction.
t
1
*
y
d
L
The range of
b
considered here is from
t
D
IfT
more or less arbitrary.
to
-
1
, the initial column cross sec.
20 and ---
tion would be a closed square.
-
The other limiting value
It can be argued that for
buckling of the stiffening plate may occur.
t
is
1
<=-.-
,
local
However, the stress in-
tensity in the stiffening plate may be well below the proportional
limit and it is hard to determine at just what value of
buckling of plate becomes critical.
t
1
t
40
1
lems.
1
T
to --
t
local
In any case, the range of
covers a considerable number of practical prob--
If J, H, and G for each of the four possible limiting conditions,
l =--,
t
W
bD4- -M
0M
- Da 20i -=a namely
1
b
D a 4; and
T
1'
b
40d4
t
1
-=
d1
125
definitions (J, equation
1
b
1
1
b
bn D a 20; are computed from their
If
k
and
q
1
125
(.26a), H, equation (26b), G, (23a)), and subox
stituted into equation (27), four distinct expressions for
a function of
t
will result.
AV as
These four equations and the
33
cross plots of
k
versas q could be used to find four optimum
curves by trial, the method being identical with that used to find
optimum curves for circular and square tubular columns.
The com.
putations involved in this approach are considerable, and a method
which depends on the two optimum curves already determined for
tubular columns and minimization criteria is used to show that the
average optimum curve already determined for square and circular
tubular columns can be used for the structural cross section under
consideration.
The derivation of the explicit form of minimization criteria
for the cross section defined in Figure 4 follows:
(27) with
q
Consider equation
replaced by f(k), being careful to note that
q n f(k)
is the same function which applies to circular and square tube cross
sections, that is, the cross plots of
k
versus
q
are independent
of the shape of the cross section considered, and depend only on
the graph representing a general solution of the characteristic e.-
quation (13).
(See Figure 5.)
Then
AV
[J + H V12G[
lJ+l
.-l) f (k) IA
k
r
1
dAV
J
S J f (k)+H
12G-- 1] + 1 ft(k)
k
f (k) H 1(12G[-
ft(k)
Jj+1)-1/2(24Gk--3
34
Let
0
and multiply both sides by
12G[ ,
k
1] + 1
Then
0 M f' (k)[(J-) 412G[k
+
k
12H
H(12G[ 1
kk
from equat ion (26a)
J n1-
H
H.
1. a
J
Therefore,
f(k) a g(k,c,D)
..
+12G[ 77k
13+1
7
k
[
f(k) -
ft(k)
1+1 1 f'(k)
. . . . . . . . .
. . . (28)
where
g(kcD)
2G [" 412G[
.
1]
+ 1
+ 12G[ 1
k
13 + 1] ..(28a)
k
and G is defined by equation (23a).
If for k = 0.5, 0.7, 0.9, 1.00,
g(kc,D) is computed using
equation (28a) for each of the four possible combinations of the
limiting vales of c M
d1
and D n
Table 1 and tabulated as follows.
,
the results can be added to
(See Appendix H.)
Table of Minimization Criteria
G(kB)
k
B
g(kcD)
1
1__
B=a TO-
2
_Dm4
100
Dm20
00
0
"T C
D
4
0
Dm 20
0
0.90
0.08995
0.08573
0,08593
0.08650
0.08552
0.08569
0.70
0.20972
0,18022
0.18061
0.18642
0.17873
0.17960
0.50
0.24961
0.19267
0.19348
0.20815
0.18799
0.19092
Table 2.
35.
It has already been shown (see Table 1) that for the f(k)
fum tions involved, a difference of 25.75% in the minimization criteria does not change the location of the optimum stiffening curve
on the graph representing a general solution of the characteristic
equation appreciably.
conclusion,
A study of Table 2 leads to the following
Since the minimization critera for the structural cross
section defined in Figure 4 lie between or slightly below the minimization critexit
for the two optimum curves obtained by trial compu-
tations for tubular columns, it is concluded that the four optimum
curves for the structural cross section in Figure 4 are very well
(See
approximated by the one average curve already determined.
Figure 5.)
A spot check of this conclusion is made for c -i D
4 , at the l]rel
\f
1
Equation (27)1with J, H,
a 0.750.
or
and G evaluated,
reduces to
AV- [.s557.o3 + 558.03VO.OO51469[ 1
k2
aV = 558.03 [ V.0051469
.
]+
]. 1
- 1]+1
.1) q
- 1] q L A1
LA1
.
.
(29)
k
If using the cross plot of
k
versus
q
for
fI or
= 0.750 and
equation (29) the kq solution associated wi. th a minimum
ution is found by trial,
&V
the minimum point thus found checks with
that given by the one average curve drawn for tubular columns.
Appendix F.)
solo
(See
36
G.
Results and Discussion
The graph of the general solution of the characteristic equation
of the simple stiffened column system defined in Figure 1 and Part II,
C, Scope, has been developed.
An average optimum stiffening curve is
superimposed on the general solution of the characteristic equation,
For columns which conform to the limitations of Part II, C, Scope,
and have cross sections which are within the ranges and of the types
considered, the optimum stiffening solution may be readily computed
using Figure 5.
Although the location of the optimum curve depends on the shape
of the cross section in a strict theoretical sense, the results of
this investigation show that the location of the optimum curves for
various cross sections does not vary appreciably.
The introduction
of an average optimum curve is reasonable, particuh rly when the
flat character of the
sidered.
AV function near the minimum point is con
The average curve presented here may well be applicable
to stiffening problems involving far more complicated structural cross
sections with two axes of symmetry.
There are two classes of problems that can be solved using the
optimum stiffening curve.
The problems of the first class are those
where an existing column is to be stiffened in order to be able to
carry more load.
The second class of problem occurs when a new col..
umn is being designed using optimum column stiffening ideas.
Class lt
As an example of problems of the first class, consider an ex-
I1
37
isting circular tube column lot long with di - 3" and t = 1".
What is the ultimate load that this column
30 x 106.)
(Steel E
can be made to support, and what is the optimum stiffening solution
which will prevent buckling failure under the ultimate load?
Compute A
I
(9 - 1) a 6.28 in 2
4(d12 00 d0 2)
i
k
and I 1
(d
164
4
4.4
3.93 in.
(81 - 1)
)
d
064
1
Check radius of gyration, to make sure initial column is in Euler
range,
0.791"
r
L-1
x 12
79
1
Assume that f
YJP.
a152
- fP
P. L.
>
.
100
O.K.
r1
= f
respect to direct stress is
= 30,000f/in 2 .
a-
#
S.F. -
Then the S.F. with
1.00.
-W
Therefore,
f
Por a S.F. P2 = 1.00 P2
T
Compute P
P.L.
p
*
~~or
Por
Pi
and f
2
2
SEI
L2
f
1
A1
1
y
x 30 x 10
6
x 3.93
- 80,900-#
1202
_ 80, 00 - 12 , 800/in2
6*28
38
or
P.L.
30,000
n 288
P~ M .
1
P
2.55
1
or
- 188,733#
Using Figure 6, find the optimum stiffening solution
= 2.33
=
q - 0.64
Optimum solution
0.654
or
P1
1
Prove
1.528
P
q = 0.64, k w 0.615
k = 0.615.
,
is a solution which increases the
strength as required.
=
-L(l.q)
S2
U =
Por
x 120 (1
"0.64)
188,733
30x10 6 x 3.93
x 120 x 0.36 x 0.040 = 0.864
2U
-n t
12
2 x 0.864 j2
S 0. 36 x Tr
2.33
2.3
Compute volume of material added for optimum solution.
k - 0.615
I2
k2 M
l
2
T (d2
10.39 in0 n 64
0.378
-
k - 0.615
2'
d 40 )
T (d 4
64
2
d 2 = 3.82 in
A 2 0. A, n
2 ?lr t = 2 1
M
1.705 x 0.41
2
A2 - A 1 a 4.40 in
AV-= (A2 - A1 ) qL a 4.40 x 0.64 x 120
&V = 338 in 3
1
0. K.
q a 0064,
39
Co mpute the volume of material added for constant cross section
st iffeninge
I2
2.33
G. 16 in
2.33 x 3.93
I
4
2
T (dd 4
64
2-
d4)
(d
~264 u2
0
1)
d2 = 3.70 in.
A
" A
2
1
A2 . A
= 2 Wr t = 2 W x 1.675 x 0.35
m
= 3.68
AV = (A2 '- Al)L = 3.68 x 120
&V = 442 in3
Compute the percentage reduction of material added by using optimum
stiffening instead of constant cross section.
% reduction
AV = 442
% reduction
AV = 23.5
-
338
%
44E2
q = 0.64, k a 0.615 is the optimum solution, run through
To show that
the design of stiffened columns represented by points a little to
either side of the optimum point.
P7--
= 0.654
Pr
(A)
q a 0.59
k = 0.595
(B)
q a 0.69
k = 0.630
Prove (A) and (B) are solutions which increase the strength as re.m
quired.
(A)
q) L (17 x 120 (0,41)(0.040) = 0.985
1=T
or
.
2U
1
I (1q
j2
[2 x 0.985j
T 0.4J.
2
3 3
0.K.
40
(B)
1
U =-f x 120 x 0.31 (0.040) = 0.744
P
or
2
P
0.
031)
L
2 = 2.33
O.K.
'-23
Compute the volume of material added for solution (A).
k
a
0.595
-3
9
k2 = 0.354
11.1 in4
d2 - 3.89 in
A
- A, W4.82 in2
V a 342 in3 > 338 in3
Compute the volume of material added for solution (B).
k = 0.630
3.43
k2 n 0.397
4
9.90 in
d2 = 3.78 in
A 2 0 A, = 4.15 in 2
V = 344,in
Therefore,
>
338 in3
q a 0.64, k w 0.615
has been shown to be the optimum
manner in which to stiffen the given column to its ultimate Po
or
188#733#.
Class 2f
Problems of the second class are those where a new column is to
be designed using optimum stiffening ideas.
These problems can be
re duced to problems of the first class by designing the initial cross
41
section first.
In designing the initial section stress A, up to the
fa, temporarily ignoring buckling failure.
The shape (square tube,
circular tube, or four angle section) will have to be decided upon
from architectural considerations or arbitrarily.
The most efficient
initial cross section will be that with the highest practicable
radius
of gyration, which is the same as saying the smallest practicable
t
ratio.
The smallest
t
that is to be permitted in any problem
1
1
must be fixed.
Some of the factors influencing the engineerts judgr*
ment in this matter are listed here.
1.
The theoretical limit for
L
values, below which local buckling
failure of the tube becomes critical.
2.
The lower limiting -g value of available manufactured sizes.
3.
The danger of damage to tubes with low
t
1
values due to handling,
erection, and fabrication is considerable.
4.
Lateral clearance considerations.
It is not desirable that any
column take up too much room by having a large d
small
L
so as to get a
ratio.
Then with the shape, Alland d
all established, the initial
cross section is completely defined, hence the problem remaining is
of the first class.
It is important to check whether the design of
the initial cross section is in the Euler range before using the
optimm stiffening curve.
The reason for desiring the smallest
t
-
practicable should be fully realized.
The percentage saving in
stiffening material is higher for less efficient heavy walled columns,
but the total volume of material used is greater.
Therefore, the
design which shows the greatest overall economy of material can be
42
achieved by stiffening the most efficient initial cross section prac-
ticable.
U
H.
Conclusions
From the results of this investigation, it may be concluded that:
1.
A readily applicable method for determining the optimum stiffening
solution for a column which is within the limitations of Part II,
C,
Scope, and has a cross section of the type, and within ranges, con.sidered, has been devised.
2.
The main source of error in the computational work is in the k
versus q cross plots.
ditional
This could be improved upon by computing ad-
curves, and by using larger scale
k
plots, but this was not considered necessary.
k
versus
q
cross
The large number of
significant figures carried in all computations is an attempt to introduce no further error over and above that inherent in the
versus
3.
q
k
cross plots.
Although savings of material are achieved by optimum stiffening
design, the cost of fabrication probably outbalances any economy of
material.
In designs where weight is a more critical factor than in
most civil engineering structures, optimum column stiffening might
well find application.
4.
The ideas successfully applied to the simplest type of column in
this investigation may be applicable to a large number of more conm
plicated column types.
(A.)
Optimum stiffening solutions for various other end conditions.
(1)
Fixed, symmetrical case.
43
(2) One end fixed, one pinned, unsymmetrical.
(3) Combinations of end conditions involving partial
fixity.
(B)
Optimum stiffening in the inelastic range of buckling
failure.
(C)
Optimum stiffening for eccentrically loaded columns, and
columns with initial curvature.
44
APPENDIX A
Notation,
L - length of the unstiffened column
r = radius of gyration of the unstiffened colUimnts cross section
=proportional limit stress intensity of the material used
f
M = resisting moment
E n Modulus of Elasticity
I = Moment of Inertia
EI= bending stiffness
P-
radius of curvature
P,
Euler buckling resistance of unstiffened column
Por= ultimate buckling resistance of stiffened column
= the moment of inertia of the unstiffened cross section
I
the total moment of inertia of the stiffened cross section
I?
kmJ
q
-
2
the ratio of the stiffened length to the total length of the
column L
AV n the volume of the stiffening material
A
- the area of the unstiffened cross section
A2 n the total area of the unstiffened cross section
K = a constant depending only on the shape of the initial unm
stiffened cross section
fY
-
the yield point stress intensity of the material used
S.F. = safety factor
U
L
a,
(1 -a q)
c
a convenient parameter
45
Notation (continued)t
d
= inside diameter of the initial tubular column
d
= outside diameter of the initial tubular column
d2 a outside diameter of the stiffening sheet for tubular columns
d
2
d
B a
ad
lo=
wall thickness of the initial tubular column
" d
d
t
a convenient constant depending on the
0
the unstiffened column
b = length of angle leg
C
convenient notation
-
b
D = T
convenient notation
ratio of
46
APPENDIX B
Discussion of Dr. Dinnikts Paper
A paper entitled Design of Columnr of Varying Cross Section, by
Dr. A. Dinnik, appears in the 1932 Transactions of the A.S.M.E., in
the Applied Mechanics Division.
a stability coefficient
In this paper, Dr. Dinnik defines
for stiffened columns identical to those
K
dealt with in this investigationg
EI
. . . . . (2) Dr. Dinnikts paper
or
L2
This is the equation which Dr. Dinnik uses to define his stability
coefficient K, in terms of the notation of this investigation.
Equation
(15) of this investigation can be written as followss
r or
1
2E
2U
oq i
2
2
2kU
1 ..i2q
Therefore, according to this investigation,
2
.
2kU
g-- q
If we solve this expression for U,
0
)
(
U W
* .
.
.
.
.
.
.
.
.
.
(30)
Using K values from Table 4 of Dr. Dinnikts paper, and the proper
associated values of
equation (30).
k
and
q
a set of roots U can be obtained from
Introduce this set of roots U, with the proper associ..
ated values of k and q, into the characteristic equation (13).
The
The com-
characteristic equation (13) should reduce to an identity.
putations here outlined follow in tabulated form.
tan U tan
kq
T -q
U = 1
k
.
.
.
.
. .
.
.
.
.
(13)
(1)
q
- 0.2
12
n
T
(2)
(3)
I
K
(4)
(5)
U=(2)(3)
tan (4)
(7)
(6)
k
(
(8)
(9)
tan (7)
(5)x(S)
(10)
1/km
0.01
0.15
15.0
3.88
0.4
1.55
48.078
0.1
0.039
0.0390
1.66
10
0.10
1.47
14.7
3.84
0.4
1.537
29.372
0.316
Q.121
0.1216
3w56
3.16
0.20
2.80
14.0
3. 74
0.4
1.498
14.100
0.450
0.169
0.1706
2.40
2.22
0.40
5.09
12.72
3.57
0.4
1.43
7.055
0.634
0.226
0.2300
1.62
1.58
0.60
6.98
11.63
3.42
0.4
1.37
4.913
0.775
0.265
0.2714
1.33
1.29
0.80
8.55
10.70
3.28
0.4
1.31
3.747
0.895
0.293
0.3017
1.13
1.12
(7)
(6).(4)
1.5
q - 0.4
0.01
0.27
27.0
5.20
0.3
1.560
92.620
0.10
2.40
24.0
4.90
0.3
1.470
9.887
0.20
4.22
21.1
4.60
0.3
1.380
0.40
6.68
16.7
4.10
0.3
0.60
8.19
13.65
3.70
0.80
9.18
11.46
3.39
0.1
0.104
0.1044
9.63
10
0.316
0.309
0.3192
3.16
3.16
5.177
0.450
0.414
0.4394
2.28
2.22
1.230
2.820
0.634
0.520
0.5726
1.1615
1.58
0.3
1.110
2.014
0.775
0.573
0.6452
1.30
1.29
0.3
1.017
1.628
0.895
0.608
0.6959
1.13
1.12
I
(1)
K
q w 0.6
(2)
(3)
/"S
(4)
(5)
U=(2).(3)
tan (4)
0.01
0.60
60.0
7.75
0.2
1.55
48.078
0.10
4.50
45.0
6.70
0.2
1.34
0.20
6.69
33.45
5.79
0.2
0.40
8.51
21.28
4.62
0.60
9.24
15.40
0.80
9.63
12.04
(6)
(7)
1.5()4)
(8)
tan (7)
(9)
(5)x(8)
(10)
1/k
0.1
0.232
0,2363
11.35
10
4.256
0,316
0.635
0,7368
3.14
3.16
1.158
2.283
0.450
0.780
0.9893
0.2
0.924
1.324
0.634
0.879
1.2072
1.60
1.58
3.93
0.2
0.786
1.001
0.775
0.915
1.2997
1,30
1.29
3.48
0.2
0.696
00835
0.895
0.935
1,3550
1.13
1.12
2.26
2,22
(7)
q - 0.8
4-(6).(4)
0.01
2.26 226.0
0.10
8.59
0.20
15.05
0.1
1.505
15.1765
85.9
9.27
0.1
0.927
9.33
46.65
6.85
0.1
0.40
9.67
24.18
4.92
0.60
9.78
16.30
0.80
9.84
12.30
0.1
0.602
0.6871
1.3525
0.316
1.171
2o3666
3.16
3.16
0.685
0.8170
0.450
1.231
2.8288
2.31
2,22
0.1
0.492
0,5360
0.634
1.248
2.9896
1.60
1.58
4.05
0.1
0.405
0.4287
0.775
1.258
3.0920
1.32
1.29
3.51
0.1
0.351
0.3662
0.895
1.258
3,0920
1.13
1.12
10.4
10
0,
49
The results of checking the stability coefficients
K
presented
in Table 4 of Dr. Dinnikts paper lead to the following discussion.
The values of (9) and
(10)
in the tables of computations just prev*
sented should be equal.
For any discrepancy which may be conskbred
too great, the value of
U
method, and the
K
may be improved by Newtonts iteration
value accordingly modified,
It would appear
from the trend of the errors indicated by the check made here, that
graphical methods were used to solve the characteristic equation while
in transcendental form, and that refinement by iteration was not
carried out, in Dinnik's paper.
Values of U taken from Table 15 of Dr. Dinnikts paper, when sub.
stituted into
K
do not give the
. . . . .
I
K
Dr. Dinnikts
values of Dr. Dinnik's Table 4.
(47)
However, the
K values of Table 4 in Dr. Dinnikts paper have been checked and
shown to be approximately correct.
Furthermore,
values of U from
Dr. Dinnikts Table 15 do not satisfy his equation (46).
inasmuch as the values of
K
Therefore,
tabulated in Table 4 of Dr. Dinnik t s
paper check when substituted into the derivation presented in this
investigation, it is suggested that equations (46) and (47), and
Table 15 of Dr. Dinnikts paper should be revised as follows.
Note that the suggested corrections are presented in terms of
Dr. Dinnikts notation.
The only difference in notation beingt
50
Dinnik
This Investigation
q L
7%
I
i
I2
L
The sign conventions are identical.
Equation (46) should be re.
vised to read
tan-A
tan U
U
1
Equation (47) should be revised to read
2kU
p
or
(1.
-2
j)A)
T
EI
2
t
Table 15 should be revised to reads
-0.z
U.4
U.6
068
0.01
1.55
1.56
1.55
1.51
0.1
1.54
1.47
1.34
0.93
0.2
1.50
1.38
1.16
0.69
0.4
1.43
1.23
0.92
0.49
0.6
1.37
1.11
0.79
0.41
0.8
1.31
1.02
0.70
0.35
Table of U Values
51
APPENDIX C
(1) Computations fbr the plot which represents the general solu
tio n of the characteristic equation (13).
tan U tan 1 -q
lr(1
-
U
I
k
. . .
. . .
0 0 0 0
q)
0
cr2U
cr
0
0
0.
.
.
.
.
.
.
.
. (13)
** **
(16)
"V
11
k
-
0,950
(1)
U
tan U
(2)
6
(3)
(4)
(5)
kU
tali- (2)
(4)/(3)
(6)
(7)
q
(8)
1 - (6)
(9)
(7)/(8)4
or
0.0000
0.0000
0.2000
0.2027
5.1928
0.1900
0.4000
0.4228
2.4895
0.6000
0.6841
0.7500
0.0000
W/2
00
1.0000
0.0000
0.0000
0.9500
1.38055
7.2660
0.8790
0.1210
0.1273
0.9605
0.3800
1.1888
3.1284
0.7578
0.2422
0.2546
0.9513
1.5386
0.5700
0.9945
1.7447
0.6357
0.3643
0.3820
0.9537
0.9361
1.1244
0.7125
0.8439
1.1844
0.5422
0.4578
0.4775
0.9587
0.8500
1.1393
0.9239
0.8075
0.7459
0.9237
0.4802
0.5198
0.5411
0.9606
0.9500
1.3984
0*7527
0.9025
0.6452
0.7149
0.4169
0.5831
0.6048
0.9641
1.0500
1.7433
0.6038
0.9975
0.5432
0.5446
0.3526
0.6474
0.6685
0.9684
1.1500
2.2345
0.4711
1.0925
0.4030
0.2872
0.7128
0.7321
0.9736
1.2500
3.0096
0.3497
1.1875
0.3364
0.2833
0.2208
0.7792
0.7958
0.9791
1.3500
4.4552
0.2363
1.2825
0.2320
041809
0.1532
0.8468
098594
0.9853
1.4000
5.7979
0.1815
1.3300
0.1795
0.1350
0.1189
0.8811
0.8913
0.9886
1.4500
8.2381
0.1278
1.3775
0.1271
0.0923
0.0845
0,9155
0,9231
0.9918
0.4403
1.5000
14.101
0.0746
1.4250
0.0745
0.0523
0.0497
0.9503
0.9549
019952
1.5500
48.078
0.0219
1.4725
0.0219
0.0149
0.0147
0.9853
0.9868
0.9985
1.5600
92.620
0.0114
1.4820
0.0114
0.0077
0.0076
0.9924
0.9931
0.9993
00
0.0000
1.4923
0.0000
0.0000
0.0000
1.0000
1.0000
1.0000
W/ 2
C')
I
k * 0.90
U
tan U
(2)
ta111
tan
U
(4)
(3)
W
tali (2)
0.0000
0.1000
0.0000
0.1003
11.0777
0.2000
0.3000
0.4000
0.5000
0.6000
0.7000
0.2027
0.3093
0.4228
0.5463
0.6841
0.8423
5.4814
3.5923
2.6279
2.0339
1.6241
1.3101
0.1800
0.2700
0.3600
0.4500
0.5400
0.6300
1.3904
1.2993
1.2072
1.1138
1.0189
0,9221
0.7500
0.9316
1.1926
0.8000
0.8500
0.9000
0.9500
1.0000
1.0500
1.1000
1.1500
1.2000
1.2500
1.3000
1.3500
1.4000
1.4500
1.5000
1.5500
1.0296
1.1393
1.2602
1.3984
1.5574
1.7433
1.9648
2.2345
2.5722
3.0096
3.6021
4.4552
5.7979
8.2381
14.101
48.078
1*0792
0.9752
0.8817
0.7946
0.7134
0.6373
0.5655
0.4972
0.4320
0.3692
0.3085
0.2494
0.1916
0.1349
0.0788
0.02311
1.5600
92.620
00
@/2
0.0000 I/2
0.0900 1.4808
(5)
(4)/(3)
(6)
(5) 2
1+(s)
(7)
q
(8)
1 - (6)
(9)
(7
F/ii
(8)(8) vpr or
1.0000
0.9427
0,0000
0.0573
0.0000
0.06366
0.9000
0.9001
7.7244
4.8122
3.3533
2.4751
1.8868
1.4637
0.8854
0.8279
0.7703
0.7122
0.6536
0.5941
0.1146
0.1721
0.2297
0.2878
0.3464
0.4059
0.1273
0.1910
0.2546
0.3183
0.3820
0.4456
0.9003
0.9011
0.9022
0,9042
0.9068
0.9109
0.6750 0.8730
1.2933
'0.5639
0.4361
0.4775
0.9133
0.7200
0.7650
0.8100
0.8550
0.9000
0.9450
0.9900
1.0350
1.0800
1.1250
1.1700
1.2150
1.2600
1.3050
1.3500
1.3950
0.8235
0.7728
0.7226
0.6714
.6197
0.5674
0.5147
0.4614
0.4078
093537
0.2992
0.2445
0.1893
0.1341
0.0787
0.02311
1.1438
1.0102
0.8921
0.7855
0.6866
0.6004
0.5199
0.4458
0.3776
0.3144
0.2557
0.2012
0.1502
0.1028
0.0583
0.0166
0.5335
0.5025
0.4715
0.4399
0.4071
0.3752
0.3421
0.3083
0.2741
0.2392
0.2036
0.1675
0.1306
0.0932
0.0551
0.0163
0.4665
0.4975
0.5285
0.5601
0.5929
0.6248
0.6579
0.6917
0.7259
0.7608
0.7964
0.8325
0.8694
0.9068
0.9449
0.9837
0.5093
0.5411
0.5730
0.6048
0.6366
0.6685
0.7002
0.7321
0.7639
0.7958
0.8276
0.8594
0.8913
0,9231
0.9549
0.9868
0.9160
0.9195
0.9223
0.9261
0.9314
0.934:7
0.9396
0.9448
0.9503
0.9560
0.9623
0.9687
0.9754
0,9823
0.9895
0.9968
0.01200 1.4040 0.01200
0.0085
0.0084
0.9916
0.9931
0.9985
0.0000
0.0000
1.0000
1.0000
1.0000
0.0000
1.4137 0.0000
00
16.453
..... .....
.- I..., . .1 -. 1 1 111.." .." .1.
I
i
11
.
I .
.
(1)
k = 0.850
U
11
1 ii6b
(2)
tan U
1.1765
0..0000
0.0000
0.2000
0.2027
0.4000
(3)
(4)
I
tann()((()
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5.8041
0.1700
0.4228
2.7826
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0.6841
0.7500
00
(2)
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(4)/(3)
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(5)
(7)
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i
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or
00
1.0000
0.0000
0.0000
0.8500
1.4002
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0,8917
0.1083
0.1273
0.8507
0.3400
1.2258
3.6052
0.7829
0.2171
0.2546
0.8527
1.7197
0.5100
1.0441
2.0472
0.6718
0.3282
0.3820
0.8592
0.9361
1.2568
0.6375
0.8987
1.4097
0.5850
0.4150
0.4775
0.8691
0.8500
1.1393
1.0327
0.7225
0.8015
1.1093
0.5259
0.4741
0.5411
0.8762
0.9500
1.3984
0.8413
0.8075
0.6994
0.8661
0.4641
0.5359
0.6048
0.8861
1.0500
1.7433
0.6749
0.8925
0.5937
0.6652
0.3995
0.6005
0.6685
0.8983
1.1500
2.2345
0.5265
0.9775
0.4846
0.4958
0.3315
0.6685
0.7321
0.9131
1.2500
3.0096
0.3909
1.0625
0.3726
0.3507
0.2596
0.7404
0.7958
0.9304
1.3500
4.4552
0.2641
1.1475
0.2582
0.2250
0.1837
0.8163
0.8594
0.9498
1.4000
5.7979
0.2029
1.1900
0.2002
0.1682
0.1440
0.8560
0.8913
0.9604
1.4500
8.2381
0.1428
1.2325
0.1418
0.1151
0.1032
0.8968
0.9231
0.9715
2
1.5000
14.101
0.0834
1.2750
0.,0832
0.0653
0.0613
0.9387
0.9549
0.9830
1.5500
48.078
0.0245
1.3175
0.0245
0.0186
0.0183
0.9817
0.9868
0.9948
1.5600
92.620
0.0127
1.3260
0.0127
0.0096
0.0095
0.9905
0.9931
0.9974
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1.3352
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Mmmm
Ai
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k = 0.8
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0.0000
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0.3000
0.4000
0.5000
(2)
tan
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500
U
k U
0.0000
0.1003
0.2027
0.3093
0.4228
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00
12.4626
6.1667
4.0413
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2.2881
0.0000
0.0800
0.1600
0.2400
0.3200
0.4000
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9.6841
1.8272
0.7000
0.7500
0.8423
0.9316
1,4840
1.3417
0.8000
1.0296
1.2141
0.8500
0.9000
0.9500
1.0000
1.0500
1.1393
1.2602
1.3984
1.5574
1.7433
1.0972
0.9919
0.8939
0.8026
0.7170
1.1000
1.9648
1.1500
1.2000
1.2500
1.3000
1.3500
1.4000
2.2345
2.5722
3.0096
3.6021
4.4552
5.7979
8.2381
1.4500
1.5000
1.5500
1.5600
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14.101
48.078
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00
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1.4907
1.4100
1.3282
1.2446
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5.5341
3.8893
2.8970
0.4800
1.0700
0.5600
0.6000
0.9778
0.9303
0.6400
0.6800
0.7200
0.7600
0.8000
0.8400
0.6362
0.8800
0.5666
0.5594
0.4860
0.4153
0.3470
0.2806
0.2156
0.9200
0.9600
1.0000
1.0400
1.0800
1.1200
0.5101
0.4524
0.3937
0.3340
0.2736
0.2124
0.1517
1.1600
0.1506
0.1298
0.1149
0.0886
0.0260
0.0135
0.0000
1.2000
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1.2480
1.2566
0.,0884
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0.0135
0.0000
0.0737
0.0210
0.0108
0,0000
0.0686
0.0206
0.0107
0.0000
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(8)
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1.0000
0.94906
0.8981
0.8470
0.7955
0.7434
0.0000
0.05094
0.1019
0.1530
0.2045
0.2.566
0.0000
0.06366
0.1273
0.1910
0.2546
0.3183
0.8000
0.8002
0.8005
0.8010
0.8032
0.8061
2.2229
0.6897
0.3103
0.3820
0.8123
1.7461
1.5505
0.6358
0.6079
0.3642
0.3921
0.4456
0.4775
0.8173
0.8212
0.8818
1.3778
0.5794
0.4206
0.5093
0.8258
0.8317
0.7814
0.7294
0.6763
0.6221
1.2231
1.0853
0.9597
0.8454
0.7406
0.5502
0.5205
0.4897
0.4590
0.4255
0.4498
0.4795
0.5103
0.5410
0.5745
0.5411
0.5730
0.6048
0.6366
0.6685
0.8313
0.8368
0.8437
0.8498
0.8594
0.6439
0.3917
0.6083
0.7002
0.8687
0.5545
0.4713
0.3937
0.3212
0.2533
0.1896
0.3567
0.3203
0.2826
0.2431
0.2021
0.1594
0.6433
0,6797
0.7175
0.7569
0.7979
0.8406
0.7321
0.7630
0.7958
0.8276
0.8594
0.8913
0.8787
0.8898
0.9016
0.9146
0.9284
0.9431
0.8851
0.9231
0.9589
0.9314
0.9794
0.9893
1.0000
0.9549
0.9868
0.9931
1.0000
0.9754
0.9925
0.9962
1.0000
orF
01
01
k w 0.750
(1)
u
tan U
(2)
1.3333
tan U
(3)
(4)
k U
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0.0000
0.0000
0.2000
0.2027
6.5777
0.1500
1.4199
0.4000
0.4228
3.1535
0.3000
0.6000
0.6841
1.9489
0.7500
0.9361
0.8500
0
(5)
(4)/(3)
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1+(5)
(7)
q
1-()
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2 U
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()r
100000
0.0000
0.0000
0.7500
9s4660
0.9045
0.0955
0.1273
0,7502
1.2637
4.2123
0.8081
0.1919
0s2546
0.7537
0.4500
1.0967
2.4371
0.7091
0.2909
0.3820
0,7615
1.4243
0.5625
0.9587
1.7044
0.6302
0.3698
0.4775
0.7745
1.1393
1.1703
0.6375
0.8637
1.3548
0.5753
0.4247
0.5411
0.7849
0.9500
1.3984
0.9534
0.7125
0.7615
1.0688
0.5166
0.4834
0.6048
0.7993
1.0500
1.7433
0.7648
047875
0.6529
0.8291
0.4533
0.5467
0.6685
0.8178
1.1500
2.2345
0.5967
0.8625
0.5380
0.6238
0.3842
0.6158
0.7321
0.8411
1.2500
3.0096
0.4430
0.9375
0.4170
0.4448
0.3079
0.6921
0.7958
0.8697
1.3500
4.4552
0.2993
1.0125
0.2908
0.2872
0.2231
0.7769
0.8594
0.9040
1.4000
5.7979
0.2300
1.0500
0.2261
0.2153
0.1772
0.8228
0..8913
0.9231
1.4500
8.2381
0.1618
1.0875
0.1594
0.1466
0.1279
0.8721
0.9231
0.9448
1.5000
14.101
0.0946
1.1250
0.0943
0.0838
0.0773
0.9227
0.9549
0.9663
1.5500
48.078
0.0277
1.1625
0.0277
0.0238
0.0232
0.9768
0.9868
0.9899
1.5600
92.620
0.0144
1.1700
0.0144
0.0123
0.0122
0*9878
019931
0.9947
00
0.0000
1.1781
0.0000
0.0000
0.0000
1.0000
1.0000
1.0000
c0n
cm
k - 0.7
(1)
U
tan U
(2)
14286
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(4)
k U
ta~
(5)
(4)/(3)
(2)
ta7
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q
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M
(8)
m
f
or
0.0000
0.0000
0
0.0000
/
1.0000
0.0000
0.0000
0.1000
0.1003
14.2433
0.0700
1.5007
214385
0.95543
0.04457
0.06366
0.2000
0.3000
0.4000
0.5000
0.2027
0.3093
0.4228
0.5463
7.0479
4.6188
3.3789
2.6150
0.1400
0.2100
0.2800
0.3500
1.4299
1.3580
1.2831
1.2055
10.2135
6.4666
4.5825
3.4442
0.9108
048661
0.8209
0.7750
0.0892
0.1339
0.1791
0.2250
0.1273
0.1910
0.2546
0,3183
0.7007
0.7011
0.7034
.0.7069
0.6000
0.7000
0..6841
0.8423
2.0883
1.6961
0.4200
0.4900
1.1242
1.0381
2*6766
2.1185
0.7280
0.6793
0.2720
0.3207
0.3820
0.4456
0.7120
0.7197
0.7500
0.8000
0.8500
09000
0.9500
1.0000
1.0500
1.100
1.1500
1.2000
1.2500
1.3000
1.3500
1.4000
1.4500
1.5000
1.5500
1.5600
27/2
0.9316
1.0296
1.1393
1.2602
1.3984
1.5574
1.7433
1.9648
2.2345
2.5722
3.0096
3.6021
4,4552
5.7979
8.2381
14.101
48,078
92.620
00
1.5335
1.3875
1.2539
1.1336
1.0216
0..9173
0.8195
0.7271
0.6393
0.5554
0.4747
0.3966
0.3207
0.2464
0.1734
0.1013
0.0297
0.0154
0.0000
0.5250
0.5600
0.5950
0.6300
0.6650
0.7000
0.7350
0.7700
0,8050
0.8400
0.8750
0.9100
0.9450
0.9800
1.0150
1.0500
1.0850
1.0920
1.0995
0.9930
0.9463
0.8976
0.8479
0.7961
0.7423
0.6855
0.628'?
0..5688
0.5070
0.4432
0.3776
0.3103
0.2416
0.1717
0.1010
0.0297
0.0154
0.0000
1.8914
1.6898
1.5085
1.3458
1.1971
1.0604
0.9326
0,8164
0.7065
0.6035
0.5065
0.4149
0.3283
0.2465
0.1692
0.0962
0.0274
0.0141
0.0000
0.6541
0.6282
0.6014
0.5737
0,5449
0.5147
0.4826
0.4495
0.4140
0.3764
0.3362
0.2932
0.2472
0.1978
0.1447
0.0878
0.0267
0.0139
0.0000
0.3459
0.3718
0.3986
0.4263
0.4551
0.4853
0.5174
0.5505
0.5860
0.6236
0.6638
0.7068
0,7528
0.8022
0.8553
0.9122
0.9733
0.9861
1.0000
0.4775
0.5093
0,5411
0,5730
0.6048
0.6366
0.6685
0.7002
0,7321
07639
0,7958
098276
0.8594
0.8913
0,9231
0.9549
0.9868
0.9931
1.0000
0.7244
0.7300
0.7366
0.7440
0,7525
0*7623
0.7740
0,7862
0.8004
0.8163
0.8341
0.8540
0.8760
0.9000
0.9265
0.9553
0.9863
0.9930
1.0000
00
2
0.7000
0.70013
Ul
.
........
...
.
k a 0.650
U8tan U
(3)
(2)
(1)
8
1
00
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0.0000
(6)
(4(5)
tano (2)
1/2
(4)/(3)
(7)
(9)
(8)
5
1..(6)
00
1.0000
0.0000
0.0000
0.6500
c
(8)
0.0000
0.0000
0.2000
0.2027
7.5900
0.1300
1.4398
11.0753
0.9172
0.0828
0,1273
0.6504
0.4000
0.4228
3.6388
0.2600
1.3026
5.0100
0.8336
0.1664
0.2546
0.6536
0.6000
0.6841
2.2489
0.3900
1.1524
2.9548
0.7471
0.2529
0.3920
0.6620
0.7500
0.9361
1.6435
0.4875
1.0242
2.1009
0.6775
0.3225
0.4775
0,6753
0.8500
1.1393
1.3504
0.5525
0.9334
1.6894
0.6282
0.3718
0.5411
0*6871
0.9500
1.3984
1.1002
0.6175
0.8331
1.3491
0.5743
0.4257
0.6048
0.7039
1.0500
1.7433
0.8825
0.6825
0.7231
1.0595
0.5144
0.4856
0.6685
0.7264
1.1500
2.2345
0.6885
0.7475
0.6030
0.8067
0.4465
0.5535
0.7321
0.7560
1.2500
3.0096
0.5112
0.8125
0.4726
0.5817
0.3678
0.6322
0.7958
0.7944
1.3500
4.4552
0.3453
0.8775
0.3325
0.3789
0.2748
0.7252
0.8594
0.8438
14000
5.7979
0.2654
0.9100
0.2594
0.2851
0.2219
0.7781
0.8913
0.8730
1.4500
8.2381
0.1868
0.9425
0.1847
0.1960
0.1639
0.8361
0.9231
0.9058
1.5000
14.101
0.1091
0.9750
0.1087
0.1115
0.1003
0.8997
0.9549
0.9422
1,5500
48.078
0.0320
1.0075
0.0320
0.0318
0.0308
0.9692
0.9868
0.9822
1.5600
92.620
0.0166
1.0140
0.0166
0.0164
0.0161
0,9839
0.9931
0.9971
TT/ 2
00
0.0000
1.0210
0.0000
0.0000
0.0000
1.0000
1.0000
1.0000
Ai
N
k - 0.6
(1)
U
tan U
(2)
tan
(3)
(4)
k U
tan 1(2)
(5)
(4)/(3)
(6)
-5
q
(7)
(8)
1-(6)
2T
(9)
(8)
1+(5)
0.0000
0.1000
0.2000
0.3000
0.4000
0.5000
0.6000
0.7000
0.7500
0.8000
0.8500
0.9000
0.9500
1.0000
1.0500
1.1000
1.1500
1.2000
1.2500
1.3000
1.3500
1.4000
1.4500
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0.1003
0.2027
0.3093
0.4228
0.5463
0.6841
0.8423
0.9316
1.0296
1.1393
1.2602
1.3984
1.5574
1.7433
1.9648
2*2345
2.5722
3.0096
3.6021
4.4552
5.7979
-8.2381
0
25.178
12.082
7.707
5.510
4.180
3,281
2.626
2.3580
2.1196
1.9043
1.7100
1.5311
1.3655
1.2110
1.0659
0.9288
0.7986
0.6744
0.5556
0.4420
0.3333
0.2294
_./
16.6168
8.2223
5.3885
3.9420
3.0508
2.4363
1.9787
1.7890
1.6188
1.4629
1.3225
1.1918
1.0702
0,9560
0.8483
0.7459
0.6480
0.5538
0.4627
0.3741
0.2875
0.2023
0.0600
0.1200
0.1800
0.-2400
0.3000
0.3600
0.4200
0.4500
0.4800
0.5100
0.5400
0.5700
0.6000
0.6300
0.6600
0.6900
0.7200
0.7500
0.7800
0.8100
0.8400
0.8700
1.5107
1.4498
1.3873
1.3224
1.2541
1.1813
1.1029
1.0611
1.0174
0.9712
0.9234
0.8727
0.8193
0.7629
0.7035
0.6409
0.5750
0.5058
0.4334
0.3580
0.2800
0.1996
1.0000
0.96179
0.9235
0.8851
0.8464
0.8069
0.7664
0.7242
0.7022
0.6794
0.6557
0.6310
0.6049
0.5773
0.5477
0.5159
0.4815
0.4440
0.4028
0.3572
0.3065
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0.1866
Pr
c
0.0000
0.03821
0.0765
0.1149
0.1536
0.1931
0.2336
0.2758
0.2978
0.3206
0.3443
0.3690
0.3951
0.4227
0.4523
0.4841
0.5185
0.5560
0.5972
0.6428
0.6935
0,7500
0.8134
0.0000
0.06366
0.1273
0.1910
0.2546
0.3183
0.3820
0.4456
0.4775
0.5093
0.5411
0.5730
0.6048
0.6366
0.6685
0.7002
0.7321
0.7639
0.7958
0.8276
0.8594
0.8913
0.9231
0.6000
0.60022
0.6009
0.6016
0.6033
0,6067
0.6115
0.6189
0.6237
0.6295
0.6363
0.6440
0.6533
0.6640
0.6766
0.6914
0.7082
0.7279
0.7504
0.7767
0.8070
0.8415
0.8811
1.5000
14.101
0.1182
0.9000
0.1177
0.1308
0.1157
0.8843
0.9549
0,9261
1.5500
48.078
0.0347
0.9300
0.0347
0.0373
0.0272
0.9728
1.5600
0.9868
92.620
0.9858
0.0180
0.9360
0.0180
0.0192
0.0188
0.9812
0,9931
0.9880
"/2
00
0.0000
0.9425
0.0000
0.00001 0.0000
1.0000
1.0000
1.0000
cmi
k - 0.575
(1)
(2)
(3)
,7391
kU
(4)
(5)
tan~ (2) (4)/(3)
(6)
-q
(7)
(8)
(9)
U
tan U
0.0000
0.0000
00
0.0000
'/2
00
1.0000
0.0000
0.0000
0.5750
0.2000
0.2027
8.5796
0.1150
1.4548
12.6504
0.9267
0.0733
0.1273
0.5758
0.4000
0.4228
4.1132
0.2300
1.3323
5.7926
0.8528
0.1472
0.2546
0.5782
0.6000
0.6841
2.5421
0.3450
1.1960
3.4666
0.7761
0.2239
0.3820
0.5861
0.7500
0.9361
1.8578
0.4313
1.0770
2.4971
0.7140
0.2860
0.4775
0.5990
0.8500
1.1393
1.5265
0.4888
0.9909
2.0272
0.6697
0.3303
0.5411
0.6104
0.9500
1.3984
1.2436
0.5463
0.8936
1.6357
0.6206
0.3794
0.6048
0.6273
1.0500
1.7433
0.9976
0.6038
0.7842
1.2988
0.5650
0.4350
0.6685
0.6507
1.1500
2.2345
0.7783
0.6613
0.6614
1.0002
0.5000
0.5000
0.7321
0.6830
1.2500
3.0096
0.5779
0.7188
0.5240
0.7290
0.4216
0.5784
0.7958
0.7268
1.3500
4.4552
0..3904
0.7763
0.3722
0.4795
0.3241
0.6759
0.8594
0.7865
1.4000
5.7979
0.3000
0.8050
0.2915
0.3621
0.2658
0.7342
0.8913
0.8237
1.4500
8.2381
0.2111
0.8338
0.2081
0.2496
0.1997
0.8003
0.9231
0.8670
1-(6)
(8)
1.5000
14,.101
0.1233
0.8625
0.1227
0.1423
0.1246
0. 8754
0.9549
0.9167
1.5500
48.078
0.0362
0.8913
0.0362
0.0406
0.0390
0.9610
0.9868
0.9739
1.5600
92.620
0.0188
0*8970
0.0188
0.0210
0.0206
0.9794.
0.9931
0.9862
00
0.0000
0.9032
0.0000
0.0000
0.0000
1.0000
1.0000
1.0000
2
0
k - 0.55
(1)
(2)
(3)
tan U
tan-18U
k U
(4)
tan'
(2)
(5)
(4)/(3)
(7)
(6)
()
1*q5
q
(9)
(8)
1..(6)
(8)
0.0000
0.0000
00
0.0000
1/2
00
1.0000
0.0000
0.0000
0.5500
0.2000
0.2027
8.9699
0.,1100
1.4598
13.2709
0.9299
0.0701
0.1273
0.5507
0.4000
0.4228
4.3003
0.2200
1.3423
6.1013
0.8592
0.1408
0.2546
0.5530
0.6000
0,6841
2.6577
0.3300
1.2109
3.6693
0.7858
0.2142
0.3820
0.5607
0.7500
0.9361
1.9423
0.4125
1.0953
2.6552
0.7264
0.2736
0.4775
0.5730
0.8500
1.1393
1.5959
0.4675
1.0111
2.1627
0.6838
063162
0.5411
0.5844
0.9500
1.3984
1.3002
0.5225
0.9152
1.7516
0.6366
0.3634
0.6048
0.6009
1.0500
1.7433
1.0430
0.5775
0.8064
1.3964
0.5827
09.4137
0.6685
0.6188
1.1500
2.2345
0.8137
0.6325
0.6830
1.0798
0.5192
0.4808
0.7321
0.6567
1.2500
3.0096
0.6041
0.6875
0.5434
0.7904
0.4415
0.5585
0.7958
0.7018
1, 3500
4.4552
0.4103
0.7425
0.3894
0.5244
0.3440
0.6560
0.8594
0.7633
1.4000
5.7979
0.3136
0.7700
0.3039
0.3947
0.2830
0.7170
0.8913
0.8044
1.4500
8.2381
0.2207
0.7975
0.2172
0,2724
0.2141
0.7859
0.9231
0.8514
1.5000
14,101
0.1289
0.8250
0.1282
0.1554
0.1345
0..8655
099549
0.9064
1.5500
48.078
0.0378
0.8525
0.0378
0.0443
0.0424
0.9576
0.9868
0.9704
1.5600
92.620
0.0196
0.8580
0.0196
0.0228
0.0223
0.9777
0.9931
0.9845
0*
0.0000
0.8639
0.0000
0.0000
0.0000
1.0000
1.0000
1.0000
2
-P
m)
I-j
k = 0.525
(1)
U
tanU
(2)
1,9048
tn
(3)
k U
(4)
an~A(2)
(5)
(6)
(5)
(4)/(3) 1+(5)
(7)
1-(6)
(9)
(8)
C?()
1
r
(
()o
00
1.0000
0.0000
0.0000
0.5250
1.4648
13.9504
0.9331
0.0669
0.1273
0.5255
0.2100
1.3524
6.4400
0.8656
0.1344
0.2546
0.5279
2.7843
0.3150
1.2260
3.8920
0.7956
0.2044
0.3820
0.5351
0.9361
2.0348
0.3938
1.1140
2.8288
0.7388
0.2612
0.4775
0.5470
0.8500
1.1393
1.6719
0.4463
1.0318
2.3118
0.6980
0.3020
0.5411
0.5581
0.9500
1.3984
1.3621
0.4988
0.9375
1.8795
0.6527
0.3473
0.6048
0.5742
1.0500
1.7433
1.0926
0.5513
0.8296
1.5048
0,6008
0.3992
0.6685
0.5972
1.1500
2.2345
0.8525
0.603q
0.7059
1.1691
0.5390
0.4610
0.7321
0.6297
1.2500
3.0096
0.6329
0.6563
0.5643
0.8598
0.4623
0.5377
0.7958
0.6757
1.3500
4.4552
0.4275
0.7088
0.4040
0.5700
0.3631
0.6369
0.8594
0.7411
1.4000
5.7979
0.3285
0.7350
0.3174
0.4318
0.3016
0.6984
0.8913
0.7836
1.4500
8.2381
0.2312
0.7613
0.2272
0.2984
0.2298
0.7702
0.9231
0.8344
0.0000
0.0000
00
0.0000
0.2000
0.2027
9.397li
0.1050
0.4000
0.4228
4.5052
0.6000
0,6841
0.7500
U/2
1.5000
14.101
0.1351
0.7875
0.1343
0.1705
0.1457
0.8543
0,9549
0,8946
1.5500
48,078
0.0396
0.8138
0.0396
0.0487
0.0464
0.9536
0.9768
0.9664
1.5600
92.620
0.0206
0.8190
0.0206
0.0252
0.0246
0.9854
0.9931
0.9822
00
0.0000
0.8247
0.0000
0.0000
0.0000
1.0000
1.0000
1.0000
2
or
k n 0.5
(1)
U
(2)
tan U
(3)
(4)
k U
(5)
tan~ (2)
(6)
(4)/(3)
tnU1+(5)
(7)
q
1.2U
(8)
.(
T.(6
(9)
(7)i
(8)
or
0.0000
.0000
00
0.0000
"/2
00
1.0000
0.0000
0.0000
0.5000
0.1000
0.1003
19.9401
0.0500
1.5207
30.414
0.9681
0.0318
0.06366
0.,50003
0.2002
0.3000
0.4000
0.5000
0.6000
0.7000
0.7500
0.8000
0.8500
0.9000
0.9500
1.0000
1.0500
1.1000
1.1500
1.2000
1.2500
1.3000
1.3500
1.4000
1.4500
1.5000
1.5500
1.5600
W/9
0.2027
0.3093
0.4228
0.5463
0.6841
0.8423
0.9316
1.0296
1.1393
1.2602
1.3984
1.5574
1.7433
L.9648
2.2345
2.5722
3.0096
3.6021
4.4552
5.7979
8.2381
14.101
48.078
92.620
9.8668
6.4662
4.7304
3.6610
2.9235
2.3745
2.1468
1.9425
1.7555
1.5870
1.4302
1.2842
1.1472
1.0179
0.8951
0.7775
0.6645
0.5552
0.4489
0.3450
0.2428
0.1418
0..0416
0.0216
0.0000
0.1000
0.1500
0.2000
0.2500
0.3000
0.3500
0.3750
0.4000
0.4250
0.4500
0.4750
0.5000
0.5250
0.5500
0.5750
0.6000
0.6250
0.6500
0,6750
0.7000
0.7250
0.7500
0.7750
0.7800
1.4699
1.4174
1.3625
1.3042
1.2412
1.1722
1.1349
1.0954
1.0530
1.0085
0.9606
0.9092
0.8530
0.7943
0.7301
0.6609
0.5865
0.5068
0.4219
0.3322
0.2382
0.1409
0.0416
0.0216
0.0000
14.699
9.449
6.813
5.217
4.137
3.349
3.026
2.739
2.478
2.241
2.022
1.818
1.6265
1.4442
1.2697
1.1015
0.9384
0.7797
0.6250
0.4746
0.3286
0.1879
0.05367
0.02769
0.0000
0.9363
0.9043
0.8720
0.8392
0.8053
0.7701
0.7516
0.7325
0.7125
0.6915
0.6691
0.6451
0.6193
0.5909
0.5594
0.5241
0.4841
0.4381
0.3846
0.3218
0.2473
0.1582
0.05094
0.02694
0.0000
0.0637
0.0957
0.1280
0.1608
0.1947
0.2299
0.2484
0.2675
0.2875
0.3085
0.3309
0.3549
0.3807
0.4091
0.4406
0.4759
0.5159
0.5619
0.6154
0.6782
0.7527
0.8418
0,9491
0.9731
1.0000
0.1273
0.1910
0.2546
0.3183
0.3820
0.4456
Q.4775
0.5093
0.5411
0.5730
0.6048
0.6366
0.6685
0.7002
0.7321
0.7639
0.7958
0.8276
0.8594
0.8913
0.9231
0.9549
0.9868
0.9931
1.0000
0.5004
0.5010
0.5028
0.5052
0.5097
0.5159
0.5202
0.5252
0.5313
0.5384
0.5471
0.5575
0.5695
0.5842
0.6018
0.6230
0.6483
0.6789
0.7161
0.7609
0.8154
0,8814
0.9618
0.9798
1.0000
1r/ 4
WAN IIIi
I
II, IAL
-. j J Ik,.,,
APPENDIX C, (a)
I.500
G4
FlGURE *5~
01 520
SQ
1~
t.
Q:7S
0.702
I.
o.7Z-
I
Q0750
0. 775
-
-
-
f
Q
01800
A 0.
K
8
0
?
4
1-*-~
08~5Q
Q~B76
V.1 --
.-
I!
-- I...--.
D3
Q00
0446
O
iB
APPENDIX C
QEras.Pk
Yxi&
(05
(3)
c f0e Cflit(n
017f
S-
2
QO
LO
'*1
TI
I.._
r
-
-t
4=
-i
~jt7
4-
-
I
-.7
J.06
090
-.
-1-
-
---
-. CI
pt
kI
Q5~
Ir
4
,
--.
0so
1.00
(Y' fe a
-
4
-
-~
04008
*
I
0480.
0.70
-i
2
04 o
ois
0.70
-7 -
oe
-----
-j0
-0.vt
I I
-4t4-A
D41O
.
_
t.
I-
K
.....
-
QLAID
.. ~-
-~
~
.~-
:.~.
-~
agso.
1T~
V..
0.10.
02 a04.
~1
1.00
a
_
_
_
_
__s
1
43 ;o
.
ito
Vo
jwl;
I
*
00L
OL "'
01"
Not-
-
- - -ib
I
-
i
lpl
09*0
-AT
-1
t
-L'
T-
~8
Ki~~
QOS
t
75
uo.~
Q.0-70
.A.
D~Iw
-
----1
-
IL
1,00
H--
+
0. (
-
I-D770
1.00 .4
U---.
-
I
-
~...---.--.---
iA
-I
-
0.9an
1.00
-,
-- ~-~---I-
E a.1 0
Q2
0,6
0.4
dr
0,S
(.9
4..
L
K.
4
4.-
d
-.
I
-..
-I
-.1'41
I
4.
4-
I..
IJ~
-~
-L
1-
;;.~I.
t
'4
I-
4
1->
I
----K~
1
4,.74
I.......I
4
7pmI
4
4,
-K.
~1-v
,II ~
J~l.
~'
J~I
~~~-~
.1
---
4
7+ ~
-.
a
-~+
~.-
10
i~..
VK.0
4
-
7)
-. 4..
Cil-4
:4
-
-.
,1
4
.4
4
-4
'1
j
I
I-
~
1
-4-
If
t
4
1
I
1
.
1..
T
4
I-
4
.1
1-..'
1
1
r-~-
4.
.4
4
1-
II
1
I
t
.
.
.
j
.
.
.i>L'..:...
174
I
>4.
..b.v
-~-..~---.--,.-.
~1~
4
L
*~.
~..
4
4
.
I
A
.4
.4
4
70
APPENDIX D
(1)
Computations for the optimum curves for circular and square
tubes,
t
It
a 1/3
1
and
d
1/202
1
For square and circular tubes
&V =
1/k
.qL(20)
80 b+u1 2... .....
For square and circular tubes
-y
[/k
t
0.0406+
.0201
d
qLA
L
/
z
.k.
1/202 1
.
. . . . . (21)
Note that equations (20) and (21) are just equation (19) with the
appropriate B value substituted in.
71
B
2
t ~1
(1)
(2)
'14
qai
'1
kj
(3)
(4)
2
V1B)\J()k
-1
(3)/k 1 -. 1 (A1)q
A 2/A1 -1
(4)
1
B +2B
AV/IA1
(2
1
0.24
0.25
0.26
0.945
0.948
0.949
80.8930
80.8987
80.9006
8.9941
8.9944
8.9945
8..5175
8.4877
8.4778
0,06468
0.06096
0.05972
0.0155
0.0152
0.0155
0.09666
0.09930
0.0348
0.0347
*
P/Por -0.975
0.36
0.35
0.920
0.918
80.8464
80.8427
8.9915
8.9913
8.7733
8.7944
0.34
0.916
80.8391
8.9911
8.8156
0.10195
0.0346
0.33
0.913
80.8336
8.9908
8.8475
0.10593
0.0349
80.7885
80.7832
80,7779
80.7744
80.7691
80.7621
80.7569
8.9882
8.9880
8.9877
8.9875
8.9872
8.9868
8.9865
9.12a8
9.1559
9.1901
9.2130
9.2476
9.2941
9.3293
0.1402
0.1445
0.1488
0.1516
0.15595
0.16176
0.16616
0.0532
0.0535
0.0536
0.0530
0.053023
0.053381
0.053171
10.0066
9.9401
9,8744
0.2508
0.2425
0.2343
0.07775
0.07760
0.07732
0.07691
*
P/Oor =0.950
0.38
0.37
0.36
0.35
0.34
0.33
0.32
0.888
0.885
0.882
0.880
0.877
0.873
0.870
-
*
P/Por =0.925
0.816
0.821
0.826
80w6659
80.6740
80.6823
8.9814
8.9819
8.9823
0.34
0.831
80.6906
8.9828
9,8096
0.2262
0.35
0.835
80.6972
8.9832
9.7583
0.2198
0.07693
0.36
0.839
80.7039
8.9835
9.7073
0.21341
0.07683
0.37
0.38
0.39
0.40
0.842
0.846
0.849
0,851
80.7090
80.7157
80.7208
80.7242
8.9838
8.9842
8.9845
8.9847
9.6695
9.6196
9.5824
9.5578
0,20868
0.20245
0.1978
0.1947
0.07721
0.076931
0.0771
0.0778
P/Por = 0.875
0.41 0.816
80.6659
8.9814
10.0066
0.2508
0.1028
0.42
0.820
80.6724
8.9818
9.9534
0.24417
0.10255
0.43
0.824
80..6790
8.9822
9.9007
0.23758
0.10216
0.44
0.45
0.46
0.828
0.830
0.833
80.6856
80.6889
80,6939
8.9825
8.9827
8.9830
9.8484
9.8225
9.7839
0.23105
0.22781
0.22298
0.10166
0.1025
0.10257
* Minimum value as indicated by computations
*
0.31
0.32
0.33
*
P/Por U0.900
72
B
m2
t
1
(.1)
q
k
P/Per
0.850
0.40
0.41
0.42
0.43
0.44
0.45
0.46
0.47
P/P
4_
(1+B) -1 (1)+k
0.73
0.779
0.783
0.788
0.791
0.795
0.798
0.800
-
(2)
80
(3)
2
(4)
Fr2
)
(3)/k -1
A 2 /A 1 .1
/AV/L
(4)
A2.1
2A+2B)A
B +2B
1
80,5975
80.6068
80.6131
80.6219
80.6257
80,6320
80.6368
80.6400
8.9776
8.9781
8.9785
8.9789
8.9792
8v9795
8.9798
8.9800
10.6139
10.5251
10.4667
10.3945
10.3517
10.2949
10.2528
10.2250
0.3267
0.3156
0.3083
0.2993
0.2939
0.2868
0.2816
0.2781
013068
0.12940
0.12949
0.12870
0.12932
0.12897
0.12954
0.13071
10.9676
10.9045
10,8266
10.7804
-10.7193
10.6589
10.5990
10.5398
0.3709
0.3630
0.3533
0.3475
0,3399
0.3323
0.3248
0.3174
0.15949
0.15972
0.158985
0.15985
0.15975
0.15950.15915
0.15870
0.825
0.43
0.44
0.45
0.46
9.47
0.48
0.49
0.50
0.750
0.754
0.759
0.762
0.766
0.770
0.774
0.778
80.5625
80,5685
80.5761
80.5806
80,5868
80e5929
80.5991
80.6053
S.9757
8.9760
8s9764
8.9767
8.9770
8.9774
8.9777
8.9780
0.51
0.780
80.6084
8.9782
10.5105
0.3138
0.160038
0.52
0.783
80.6131
8.9785
10.4667
0.3083
0.160316
80.5300
80.5373
80.5446
80.5506
80.5565
80.5610
80.5670
80.5715
80.5761
80.5806
8.9739
8.9743
8.9747
8,9750
8.9753
8.9756
8.9759
8.9762
8.9764
8.9767
11.3267
11.2432
11.1608
11.0956
11.0312
10.9834
10.9201
10.8732
10.8266
10.7804
0.4158
0.4054
0.3951
0.3869
0.3789
0.3729
0.3650
0,3591
0.3533
0*3475
0.191268
0.190538
0.189648
0.189581
0.189450
0.190178
0.189800
0.190323
0.190782
0.191125
P/Per= 0.800
0.46
0.47
0.48
0.49
0.50
0.51
0.52
0.53
0,54
0.55
0.728
0.733
0.738
0.742
0.746
0.749
0.753
0.756
0.759
0,762
x
* Minimum value indicated by computations
73
2
B
t=1
(1)
(2)
(3)
(4)
4
q
k
P/Por
0.775
0.49
0.50
0.51
0.52
0.53
0.54
0.55
0.56
(4)
(1+B) -1 (1)+k'
0.705
0.710
0.714
0.718
0.722
0.726
0.730
0.733
X 2/A1 -l
80.4970
80.5041
80.5098
80.5155
80.5213
80.5271
80.6329
80.5373
Vfi)
(3)/k -1
8.9720
89724
8.9727
8.9730
8.9734
8.9737
8.9740
8.9743
11.7262
11.6371
11.5668
11.4972
11.4285
11.3604
11.2931
11.2432
V/IAl
A2
-l (3 +2B3
-1) q
0.4657
0.4546
0.4458
0.4371
0.4285
0.4200
0.4116
0.4054
0.228193
0.227300
0.227358
0.227292
0.227105
0.22680
0.22638
0.227024
0.57
0.736
80.5417
8.9745
11.1936
0.3992
0,227544
0.58
0.739
80.5461
849748
11.1445
0.3930
0.22794
80.4624
80.4679
80.4747
80.4802
80.4858
8,9701
8.9704
8.9708
8.9711
8.9714
12.1913
12.1146
12.0200
11.9453
11.8714
0.5239
0.5143
0.5025
0,4931
0.4839
0.26719
0.26744
0.26632
0.26627
0.266145
0.26588
0.51
0.52
0.53
0.54
0.55
0,680
0.684
0.689
0.693
09.697
0.56
0.701
80,4814
8.9719
11.7984
0.4748
0.57
0.703
80.4942
8.9721
11.7623
0.4702
0.26801
0.58
0.706
80.4984
8.9723
11.7083
0.4635
0.26883
P/Por
0.725
0.53
0.54
0.55
0.56
0.57
0.58
0.59
0.60
0.655
0.660
0.664
0.669
0.672
0.677
0.680
0,683
80.4290
80.4356
80.4409
80.4476
80.4516
80.4583
80,4624
80.4665
8.9682
8.9686
8,9689
8.9693
8.9695
8.9699
8w.9701
8.9703
12.6919
12.5887
12.5073
12.4070
12.3474
12.2494
12.1913
12.1336
0.5864
0.5735
0.5634
0.5508
0.5434
0.5311
0.5239
0.5167
0,310792
0.309690
-0.309870
0.308448
0-309738
0.308038
0.309101
0.31002
0.61
0.62
0.687
0.690
80.4720
80.4761
8.9706
8.9709
12.0576
12.0013
0.5072
0.5001
0.30939
0.31006
* Minimum value indicated by computations
*
P/Por a0.750
,-4
74
B
2
t
1
(1)
q
k
4
(1+B) .1
(2)
(1)+k2
(3)
(4)
q
(3)/k -1
A 2/Al -1
/1
B2+2B
A2
(-1)
A
0.6182
0.6048
0.5943
0.5838
0.5761
0.35237
0.35078
0.35064
0.35028
0.35142
(4)2
-
1
1
P/Porn 0.700
0.57
0.58
0.59
0.60
0.61
0,643
0.648
0.652
0.656
0.659
P/Pcr
8094134
80w4199
8094251
80.4303
80.4343
8.9674
8.9677
8o9680
8.9683"
8.9685
12.9461
12.8390
12.7546
12.6711
12.6092
0.675
0.59
.0.619
80.3832
8.9657
13.4841
0.6855
0.404445
0.60
0.61
0.62
0.63
0.64
0.623
0.627
0,631
0.634
0.638
80.3881
80.3931
80.3982
80.4020
80.4070
8.9659
8.9662
8.9665
8.9667
8.9670
13.3914
13.3001
13.2099
13.1430
13.0548
0.6739
0.6625
0.6512
0.6428
0.6318
0.404340
0.404125
0.403744
0.404964
0.404352
0.65
0.66
0.640
0.643
80.4096
80.4134
8.9671
8.9674
13.0110
12.9461
0.6263
0.6182
0.407095
0,408012
0.67
0.68
0.647
0.650
80.4186
80.4225
8.9676
8.9679
12.8602
12.7967
0.6075
0.5995
0.407025
0.407660
80.3352
80.3411
80.3469
80.3516
80.3564
80.3612
80o3660
80.3709
80.3733
80.3770
8.9630
8.9633
8.9636
8.9639
8.9642
8.9644
8.9647
8.9650
8.9651
8.9655
14,4801
14.3481
14.2183
14.1161
14.0164
13.9158
13,8176
13.7208
13.6728
13.6014
0.8100
0.7933
0.7772
0.7645
0.7519
0.7394
0.7272
0.7151
0.7091
0.7001
0.4617
0.460114
0.458548
0.4587
0.458659
0.458428
0.458136
0.457664
0.460915
0.462066
P/Por
0.57
0.58
0.59
0.60
0.61
0.62
0;63
0.64
0.65
0.66
0.650
0.579
0.584
0,589
0.593
.597
0.601
0.605
0.609
0.611
0.614
* Minimum value indicated by computations
q
75
=2
t
1
(1)
q
k
2
(1+B) .1 (1)+k'
P/Por a0.625
0.61 0.566
0.62 0.570
0.63 0.575
0.64
0.579
0.65
0.66
0.585
0,587
0.67
0,590
0.68
0.593
0.69
0.70
(2)
80.3204
80.3249
80.3306
80,3352
80.3399
80.3446
80.3481
(3)
V(2)
8. 9622
8.9624
8.9627
809630
0.596
80.3516
80.3552
8.9633
8.9635
8.9637
8,9639
8.9641
0.599
80.3588
8.9643
P/Por a0.600
0.65 0.553
0.66 0.558
0.67
0.561
0.68
0.565
0.69
0,569
0.70
0.571
P/Por =0.575
0.63 0.516
0.64
0.65
0.520
0.525
0.66
0.530
0.67
0.533
0.68
0.69
0.537
0.540
0.70
0.71
0.72
0.543
0.546
0.550
P/Pora0.550
0.67 0.505
0.68 0.510
0.69 0.513
0.70 0.517
80.3058
8.9614
80.3114
8.9617
8.96185
8.8621
80.3147
80.3192
80.3237
80.3260
80.2662
80.2704
80.2756
80.2809
80.2840
80*2884
80.2916
80.2948
80.2981
80.3025
(4)
(3)/k -1
14.8372
14.7235
14.5873
14.4801
14.3744
14.2700
14.1927
14.1161
A 2/A.1
bV/L%
(4)A2
-1%( 2 1) q
1
B +2B
0.8542
0.8404
0.8234
0.8100
0.7968
0.7645
0.7550
0.51986
0.52095
0.7456
0.52192
14.9747
0.9006
0.8825
0.8718
14.0404
13.9654
15.2050
15.0603
14.8621
0.8577
8.96235
8.9625
14.7510
14.6961
0,8438
0.8370
0.585390
0.582245
0.584106
0.583236
0.582222
0.585900
8,9591
8.9594
809597
8.9600
16.3625
1.0453
1.0287
0.658539
0.658368
1.0082
0.9945
0.655330
0.656370
8.9601
15.8106
15.6860
15.5937
15.5022
15.4119
15. 2930
0.9763
0.9607
0.9492
0.9377
0.654121
0.653276
0.9264
0,657744
009116
0.656352
1.0924
0.731908
0.728076
0.729951
0.728770
0.734688
809604
8.9606
8.9607
8.9609
8.9612
16.2296
16.0660
15.9056
80*2601
80.2632
8.9590
80,2673
8.9592
16.3292
8.9594
8.9595
16.2296
1.0411
1.0287
16.1637
1.0204
0.520
80.2704
0.72
0.522
80.2725
0.51840
0.7740
0.7837
16.7396
16.5662
16.4639
0.71
0.51874
0.51792
0.51724
0.51858
8.9585
8,9588
80.2550
0.52106
0.52104
* Minimum value indicated by computations
1.0707
1.0539
0.654948
0,65639
0.730377
*
B
76
t
1
B
(1)
q
k
(1+B)4 -1
(2)
(3)
(1)+k2
(4)
A 2/A1
(3)/k -1
.0406
.1
&V/Ikl
-1 (A -1)
B+2B
1
0.0291
0.0282
0.942
0.945
0.25
0.26
0.27
0.9280
0.9336
0.96333
0.96623
0.02264
0.02246
0.1264
0.1174-
0.948
0.9393
0.96918
0.02234
0.1111
0.0278
0.949
0.950
0.9412
0.9431
0.97016
0.97113
0.02229
0.02224
0.1089
0.1065
0.0283
0.0297
P/Por= 0.950
0.36
0.920
0.35
0.918
0.34
0.916
0.8870
0.8833
0.9797
0.946181
0.93984
0.93792
0.02370
0.02379
0.02393
0.1791
0.1836
0.1905
0.6448
0.6426
0.6477
0.33
0.9742
0.93499
0.02408
0.1980
0.6534
0.913
0.39
0.889
0.8309
0,91154
0.02535
0.2612
001019
0.38
0.888
0.8291
0.91055
0.02539
0.2632
0.1000
0.37
0.36
0.35
0.34
0.885
0.882
0.880
0.877
0.8288
0.8185
0.8150
0.8097
0.90763
0.90471
0.90277
0.89983
0.02557
0.02574
0,02587
0.02604
0.2721
0.2806
0.2871
0 .2950
0.1006
0.1010
0.1005
0.1003
0.33
0.873
0,8027
0.89594
0.02627
0.3070
0.1013
0.32
0.870
0.7975
0.89303
0.02647
0.3169
0.1014
P/Per
0.900
0.31
0.816
0.7065
0.84054
0.03007
0.4960
0.1538
0.32
0.33
0.34
0.35
0.821
0.826
0.831
0.835
0.7146
0.7229
0.7312
0.7378
0.84534
0.85024
0.8551
0.85895
0.02964
0.02934
0.02900
0.02868
0.4746
0.4597
0.4428
0.4269
0.1519
0.1517
0.1506
0.1494
0.36
0.839
0.7445
0.86284
0.02841
0.4134
0.1488
0,37
0.38
0.39
0.40
0..842
0.846
0.849
0.851
0.7496
0.7563
0.7614
0.7648
0.86579
0.86966
0.87258
0.87453
0.02825
0.02796
0.02777
0.02764
0,4055
0.3910
0.3816
0.3751
0.1500
0.1486
0.1488
0.1500
P/Por
*
0.925
*
cr
0.875
0.41
0.816
0.7065
0.84054
0.03007
0.4960
0.2034
0.42
0.43
0.44
0.45
0.46
0.47
0.820
0.824
0.828
0.830
0.833
0.835
0.7130
0.7196
0.7262
0.7295
0.7345
0.7378
0.84439
0.84829
0.85217
0.85411
0.85703
0.85895
0.02974
0.02947
0.02919
0.02904
0.02884
0.02808
0.4796
0.4662
0.4522
0.4448
0.4348
0.4269
0.2014
0.2005
0.1990
0.2002
0.2000
0.2006
* Minimum value indicated by computations
*
P/P o
*
0.23
0.24
*
P/P rx 0.975
77
t
n
B
q
k
(2)
(1+B) 4 1 (1)+k
.0406
(3)
4(2)
(4)
(3)/k -1
A 2/ /A
1j. Av/L1
a74-
*
(-
1) q
.-
B2
P/Por* 0.850
0.40
0.773
0.6381
0.79881
0.03338
0.6607
0.2643
0.41
0.42
0.779
0.783
0.6474
0.6537
0.80461
0.80852
0.03287
0.05259
0.6353
0.6214
0.2605
0.2610
0.43
0.44
0.45
0.46
0.788
0.791
0.795
0.798
0.6615
0.6663
0.6726
0.6774
0.81333
0.81627
0.82012
0.82304
0.03214
0.03194
0.03159
0.03137
0.5990
0.5891
0.5716
0.5607
0.2576
0.2592
0.2572
0.2579
0.47
0.800
0.6806
0.82498
0.03122
0.5532
0.2600
0.03456
0.03507
0.7642
0.7448
0.3286
0.3277
0.3258
*
(1)
0.43
0.44
0.750
0.754
0.6031
0.6091
0.77660
0.78045
0.45
0.759
0.6167
0.78530
0.03465
0.7239
0.46
0.762
0.6212
0.78816
0.03433
0.7080
0.3257
0.47
0.766
0.6274
0.79209
0.03406
0.6945
0.3264
0.48
0.770
0.6335
0.79593
0.03367
0.6751
0.3240
0.49
0.50
0.774
0.778
0.6397
0.6459
0.79981
0.80368
0.03334
0,03300
0.6587
0.6418
0.3228
0.3209
0.51
0.52
0.53
0.54
0.780
0.783
0.786
0.788
0.6490
0.6537
0.6584
0.6615
0.80561
0.80852
0.81142
0.81333
0.03283
0.03259
0.03234
0.03214
0.6333
0.6214
0.6090
0.5990
0.3230
0.3231
0.3228
0.3235
or
0.800
0.46
0.47
0.728
0.783
0.5706
0.5779
0.75538
0.76020
0.03760
0.03710
0.8706
0.8458
0.4005
0.3975
0.48
0.49
0.738
0.742
0.5852
0.5912
0.76498
0.76890
0.03655
0.03625
0.8184
0.8035
0.3928
0,3937
0.50
0.746
0.5971.
0.77272
0.03581
0.7816
0.3908
0.51
0.749
0.6016
0.77563
0.03555
097687
0.3920
0.52
0.53
0.753
0.756
0.6076
0.6121
0*77949
0.78237
0.03517
0.03488
0.7498
0.7353
0.3899
0.3897
0.54
0.759
0.6167
0.78530
0.03465
0.7239
0.3909
0.55
0.762
0.6212
0.78816
0.03433
0.7080
0..3894
0.56
0.57
0.58
0.59
0.764
0.767
0.770
0.772
0.6243
0.6289
0.6335
0.6366
0.79013
0.79303
0.79593
0.79787
0.03420
0.03393
0.03367
0.03351
0.7015
0.6881
0,6751
0.6672
0.3928
0.3922
0.3916
0.3936
* Minimum value indicated by computations
*
P/P o
*
P/P r- 0.825
f7 ?
t
l
B
202
B=1
(1)
(1+B) 4 -1
k
(1)+k2
(3)
2
(
(4)
4(2-(3)k
(3)/k -1
A V/IAI
A
(4)
B +2B
0.0406
P/Por= 0.775
0*49
0.705
0.710
0.50
0.714
0.51
0.718
0.52
0.722
0.53
0.54
0.726
0.55
0.730
0.733
0.56
0.736
0.57
0.739
0.58
0.741
0.59
0.60
0.743
A2
.1
I1
0.5376
0.5447
0.5504
0.5561
0.5619
0.5677
0.5735
0.5779
0.5823
0.5867
0.5897
0.5926
0.73321
0.73804
0.74189
0.74572
0.74960
0.75346
0.75730
0.76020
0.76309
0.76596
0.76792
0.76981
0.04001
0.03949
0.03906
0603860
0.03822
0.03782
0.03739
0.03710
0.03680
0.03648
0.03632
0.03608
0.9905
0.9647
0.9433
0.9204
0.9015
0.8816
0.8602
0.8458
0.8308
0.8149
0.8070
0.7950
0.4853
0.4824
0.4811
0.4786
0.4778
0.4761
0.4731
0.4736
0.4736
0.4726
0.4761
0.4770
0.70923
0.71309
0.71784
0.72166
0.72553
0.72938
0.73130
0.04298
0.04252
0.04185
0.04135
0.04093
0,04048
0.04025
1.1383
1.1154
1.0821
1.0572
1.0363
1.0139
1.0025
0.5805
0.5800
0.5735
0.5709
0.5700
0.5678
0.5714
P/Por
0.51
0.52
0.53
0.54
0.55
0.56
0.57
= 0.750
0.680
0.684
0.689
0.693
0.697
0.701
0.703
0.5030
0.5085
0.5153
0.5208
0.5264
0.5320
0.5348
0.58
0.706
0.5390
0.73410
0603980
0.9801
0.5685
0.59
0.709
0.5433
0.73709
0.03961
0.9706
0.5727
0.60
0.711
0.5461
0.43989
0.03935
0.9577
0.5746
P/Por
0.725
0.53
0.54
0.55
0.56
0.655
0.660
0.664
0.669
0.4696
0.4762
0.4815
0.4882
0.68527
0.69007
0.69390
0.69871
0.04621
0.04556
0.04503
0.04440
1.2990
1.2667
1.2403
1.2090
0.6885
0.6840
0.6821
0.6770
0.57
0.58
0.59
0.60
0.61
0.62
0.63
0.64
0.672
0.677
0.680
0.683
0.687
0.690
0.692
0.695
0.4922
0.4989
0.5030
0.5071
0.5126
0.5167
0.5195
0.5236
0.70157
0.70633
0.70923
0.71211
0.71596
0.71882
0.72076
0.72360
0.04400
0.04332
0.04298
0.04262
0.04215
0.04176
0.04156
0.04115
1.1891
1.1552
1.1383
1.1204
1.0970
1.0776
1.0677
1.0473
0.6778
0.6700
0.6716
0.6722
0.6692
0.6681
0.6726
0.6703
0.65
0.66
0.698
0.700
0.5278
0.5306
0.72650
0.72842
0.04083
0.04060
1.0313
1.0199
0.6703
0.6731
*
q
(2)
79
t
1
B
10
(1)
(2)
(3)
472-(4
q
k
(1+B) -1 (1)+k
(4)
A2
A2/L
q(2)
(3)/k -1
(4
B +2B
'cro 0.700
0.57
0.643
0.58
0.648
0.59
0.652
0.60
0.656
0.61
0.659
0.62
0.661
1
(q
1
P
Pl/Por
0.4540
0.4605
0.4657
0.4709
0.4749
0.4775
0.67380
0.67860
0.68242
0.68622
0.68913
0.69101
0.04790
0.04722
0.04665
0.04606
0.04572
0.04540
1.3831
1.3493
1-03209
1.2915
1.2746
1.2587
0.7883
0.7826
0.7793
0.7749
0.7775
0.7804
0.4238
0.4287
0.65100
0.65475
0.05169
0.05096
1.5716
1.5353
0.9272
0.9212
0.675
0.59
0.60
0.619
0.623
0.61
0.627
0.4337
0.65856
0,05033
1.5040
0.9174
0.62
0.631
0.4388
0.66242
0.04979
1.4771
0.9158
0.63
0.634
0.4426
0.66528
0.04933
1.4542
0.9161
0.64
0.638
0.4476
0.66903
0.04863
1.4194
0.9084
0.65
0.66
0.640
0.643
0.4502
0.4540
0.67097
0.67580
0.04839
0.04790
1.4075
1.3831
0.9149
0.9128
0.67
0.68
0.69
0.647
0.650
0.651
0.4592
0.4631
0.4644
0.67764
0.68051
0.68147
0.04735
0.04693
0.04680
1.3557
1.3348
1.3284
0.9083
0.9077
0.9166
0.70
0.653
0.4670
0.68337
0.04650
1.3134
0.9194
1.0966
1.0910
1.0793
P1/Pcr= 0.650
0.57
0.58
0.59
0.579
0.584
0.589
0.3758
0.3817
0.3875
0.61303
0.61782
0.62250
0.05877
0.05791
0.05687
1.9239
1.8811
1.8294
0.60
0.593
0.3922
0.62626
0.05608
1.7900
1.0740
0.61
0.62
0.597
0.601
0.3970
0.4018
0.63008
0.63388
0.05541
0.05470
1.7567
1.7214
1.0716
1.0673
0.63
0.605
0.4066
0.63765
0.05396
1.6846
1.0613
0.64
0.65
0.609
0.611
0.4115
0.4139
0.64148
0.64335
0.05333
0.05294
1.6532
1.6338
1.0580
1.0620
0.66
0.614
0.4176
0.64622
0.05247
1.6104
1.0629
0.67
0.617
0.4213
0.64908
0.05199
1.5866
1.0630
0.68
0.619
0.4238
0.65100
0.05169
1.5716
1.0687
80
1
t
1
1
B 100
(1)
(2)
(3)
(4)
A2
/
1
q
k
(1+B) -.
1
(1)+k
\J(2)
1)q
(3)/k-1
B2+2B
1
P /Por= 0.625
0.61
0.62
0.63
0.64
0.65
0.66
0.67
0.68
0.69
0.70
0.566
0.570
0.575
0.579
0.583
0.587
0.590
0.593
0.596
0.599
0.3610
0.3655
0.3712
0.3758
0.3805
0.3852
0.3887
0.3922
0.3958
0.5994
0.60083
0.60457
0.60926
0.61303
0.61685
0.62064
0.62346
0.62626
0.62913
0.63198
0.06153
0.06064
0.05958
0.05877
0.05806
0.05731
0.05671
0.05608
0.05558
0.05505
2.0612
2.0169
1.9642
1.9349
1.8886
1.8512
1.8214
1.7900
1.7652
1.7388'
1.2573
1.2505
1.2374
1.2313
1.2276
1.2218
1.2203
1.2172
1.2180
1.2172
0.71
0.72
0.601
0.603
0.4018
0.4042
0.63388
0.63577
0.05470
0.05434
1.7214
1.7035
1.2222
1.2265
0.73
0.606
0.4078
0.63859
0.05377
1.6751
1.2228
0.74
0.609
0.4115
0.64148
0.05333
1.6532
1.2234
Pl/Pr = 0.600
I r
0.65
0.553
0.3464
0.58856
0.06430
2.1990
1.4294
0.66
0,558
0.3520
0.59330
0.06326
2.1473
1.4172
0.67
0.561
0.3555
0.59624
0.06281
2.1249
1.4237
0.68
0.565
0.3598
0.59983
0.06164
2.0667
1.4054
0.69
0.70
0.569
0.571
0.3643
0.3666
0.60357
0.60548
0.06075
0.06038
2.0224
2.0040
1.3955
1.4028
0.71
0.573
0.3689
0.60737
0.05998
1.9841
1.4087
0.72
0.576
0.3724
0.61025
0.05946
1.9582
1.4099
0.73
0.74
0.75
0.579
0.580
0.581
0.3758
0.3770
0.3782
0.61303
0.61400
0.61498
0.05877
0.05862
0.05848
1.9239
1.9164
1.9095
1.4044
1.4181
1.4321
81
t
1
B
10
(1)
(2)
(3)
(4)
A2
-1
q
k
4
(1+B) -1
2
(3/A
(3)/k-1
(1) +k
(4-
hYIA
-1 (
21
-1)
q
P /'cra 0.575
0.71
0.72
0.73
0.74
P 'r
00540
0.543
0.546
0.550
0.551
0.553
0.70
0.71
0.-72
0.73
0.74
0.522
0.525
0.527
0.68
0069
0*07108
0..06983
2.5363
204741
1.6486
1.6329
0.06893
2.4294
1.6277
0.3290
0.3322
0.3354
0.57359
2.3896
2.3507
0.57914
0.06813
0.06735
0.06655
0.3387
0.3431
0.58198
0.58575
0.06589
0.06500
0.3442
0.3464
0.58669
0.58856
0.06477
0.54369
0.07661
0.07521
0.07442
0.07328
0.3215
0.3246
0.57637
0.06430
2.3109
2. 2781
2.2338
2.2224
2.1990
1.6249
1.6220
1.6176
1.6175
106083
1.6223
1.6273
m0.550
0.505
0.510
0.513
0.517
00520
0.67
0.56232
0.56701
0.56974
0.3162
*
0.67
0.68
0.69
0.070
0.525
0.530
00533
0.537
0.2956
0.3007
0. 3038
0.3079
0.3110
0 3131
0.54836
0.55118
0;6 55489
0.55767
0.55955
0.3162
0.56232
0.3183
0.56418
2.8114
2.7418
2.7025
1.8836
2.6458
2.6040
1.8521
0.07244
0.07193
0.07108
0.07055
2.5786
2. 5363
2.5100
1.8644
1.8647
1.8488
1.8566
1.8515
1.8574
*
0.65
0.66
Zt
0.
c
-_j
7t2
'2
--
--'44. u-b
-T4-
K.-7
.uj
-.1
4
f
-'-
-~it5PU 7
F
XiL4
Ii'
-4 F
F
F
i
4
t
-
44
F.,
Y
-4
-J
-F'
7-I
-
J
~i1
-F
F
i4~r
F
4-F.
'4- 1
-
-
-;
[7
'4.,
1. ,T
F
j2~V
'F
FF
1 17-
Iii
ii]
7Y7 PT.
L~t
~F;4-i-
-F
4
-
'F
''1
'a
VI.
1---'-;
i:
1i21.221
4-
4-
i-F--F
3-
-1
-4.
H-!~4I
4-
.27
I 1-t1
-fl-i-I
14-
K
44;. -4--f---FF4
4+
l
~iii:TiIPi
j
4'~7ti
17 7j-77
' 4i k7
'4~ v~ J
t
F.~FIF-
4-7
77 :4*77'
~
:--7
4FF
.4
F
-FT
*T
F
4Ft~-T
t4
7
--
47
444
,
At
'
F' F
F
--~.
F'
-F.-L
-
F
4
, ,
F.
F>
F
4
I
FFF
1 4
IF
[F.:t
V7
r
.''..4:7.
' ~F - F
-
F7,
'4
SI,
F, ;
7I~
17
-F
F .4. F
4~F.F4~
'.4.
'--4.'
1.
F
"'[ii
Ft
I44
-44
1-., - I-:
-
F
7t1'J F
- F -F 4*'
:
''F~''F
I- 1 1 t
71
_
1
ZTI
1
4-
:4-I
'
'F'
1.4
0o-
,I~
-]
ac
83
APPENDIX E
Algebra of method for checking validity of equation (26)
A
J+
J + H
where
H *
B
4c
= 1
0
0
.
0
0
(2D - 1)
B
or
d
1
d
1
t
. . . . . . . .
H
.
2
0
[ j .- ..J
-
1
1
d
o
g1
1c-
B
1/2" -2(B1+ 1) '
2(BB+ 1)
b
but special condition
D
D
1
-
c0
.0
0
0
0
0
0
(26)
d4
d 1d
*
1
o
d
0
d1
D W B +1
Dc 2
402 =
(B+1)
B
4
. . . .
(26a)
(2D - 1)
4c
.
0
2
e8Dc
- 4c -
=
0
k12G
k2
B
2(B + 11
B
2(B+ 1)
2.
B
2(B + 1)
2
B
TB + 1)r
B
4(B + 1)
0
0
0.
0 .
(26b)
84
B2
B2
4c2 (2D - 1)-
2B +2
(B + 1)
M1) B
4 2(2D
(2B + 2
B)
-
B(B + 2)
(B + 1)
-
(B + 1)
G defined by equation
2(1-2cD) 03
(23a)
.2c(1
1 o 2cD n 1 - 2 r Z(B+B17
2B
B+
- 2cD)
3
0o
B +1
)4
(1..
G - 1 Ci
12B+f
A2
3
- (I - 2c)4] - 6(l .. 2cD) c (,.C)2
G = 11
B2
(1.
1 ) + v[1
(B+1)
B
S
)4][
B+1
A1
12
.
+1
k
B(B + 2)
(B + 1)
multiply top and bottom by (B
A2
[2B(B+1) ~ B2 - (B+1)
2
+ 1) 2
- 1
) + q[(B+1)
10
+ (B+1)4
i1
2
B 2+ 2B
2
-I4
0 1 + 1/k v(B+1)
. 1] l - k2
+ (B+1)
42';
k2
B2 + 2B
A
r~2
1
1/k
(B+1)
2
B + 2B
Equation (26) checks*
+ k2
*
#
0 0 0*
0
*
(18)
85
APPENDIX F
Computations for spot check of the conclusions drawn from Table II.
&V - 558.03
[
\ 0.0051469
(
1
.
-
k
1
)
+ 1
1
3
qL A
*.(29)
1
D m4
k
q
(A)
558.03 (A)
AV
Trw
1r- 0.750
0.53
0.689
0.002922
106306
0.8642
0.54
0.69,3
0.002781
1.5519
0.8380
0*55
0.697
0.002740
1.5290
0.8410
0.56
0.701
0.002660
1.4844
0.8312
0.:57
0.703
0.002630
1.4676
0.8365
0.58
0.706
0.002586
1.4430
0.8370
0.59
0.709
0.002543
1.4191
0.8373
0.60
0.711
0.002514
1.4029
0.8417
I
(A) -
1
0.0051469
(kya
+ 1
)
Notet
- 1 ]
*
cr
W
qq-wl-
86
APPENDIX G
Computation of minimization criteria for circular and square tubes*
(See Table I.)
G(kB) o
k[(1+B) 4
14
(1+B) - 1(22a)
k n 1.00
G(k,B) =
(B+l) 2 k 2(1+B)401+k2 1/2
B I 2
1
(1+B) 4 - 1
4
G(kB) = 0
k = 1.00
G(kB) =
k =*90
B
0
B = 2
G(kB) =
1 Eo.9 (80.81) - 9 x 0.81 x 8.98944]
G(kB) a 0.08995
k = 0.90
B
-
[(2+B) 44(B+1)2 (B+)41/2
1 + k2 ]
G(kB) - 24.6281 [0.9 (0.85060401 - 1.0201 x 0.81 (0.92228)]
= 24.6281 L0.765543609 - 0.762062441]
24.6281 [Q.00348117]
G(kB)=
0.08573
k = 0.70
B = 2
G(kB)-
[0.7 (80.49) - 9 x 0.49 x 8.9716]
[
[56.34300 - 39.564756]
-
G(kB)m
( 16 778244]
0.20972
87
B m
k a 0.70
16
G(k,B) - 24.6281 [ 0.7 (0.53060401)- 1.0201 x 0.49 (0.72843)]
= 24.6281
f
0.3714228 - 0.3641050]
= 24.6281
f
0.0073178]
G(kB) n 0.18022
k n 0.50
B = 2
G(kB) -
[ 0.5 (80.25) . 9 x 0.25 x 8.95824)]
40.12500 - 20.15604]
80
G(kB)
-
k = 0.50
[19*96896]
0.24961
B
G(k,B) n 24.6281 [0.5 (0.29060401) - 1.0201 x 0.25 (0.53908)]
= 24.6281 [0.145302005 - 0.137478877]
= 24.6281 [0.00782313]
G(kB) a
0.19267
8e
APPENDIX H
Computations for minimization criteria for structural cross section.
(See Table I.)
+ 12G
1
g(kc,D)
11 + 1] . . (28a)
kk
1
12
4 6c(1..2cD)(1-c) 2-2(1*-20D)c 3 .20(1-2cD)
1~-- (1--2c) .-
t
(1)
C
1
t
a=
(2)
4
G
20
G = 1
[0.18549371
D.
4
G = 1
tO.0051469]
D
20
G =
[0.025 1289]
D
-
G
D
*
(23a)
T2 [0.0457937]
=1
t
(3)
al=
C
W3
= 1
t
(4)
-=
k = 1.00
(1)
g(k,cD) = 0
(2)
g(k,c,D) = 0
(3)
g(kcD) a 0
(4)
g(k,cD) a 0
k = 0.90
3
(1)
[0.0457937 x 0.234567 + 1 -
g(k,c,D) =
Q.729
[1.01074169 - 1.0052 - - am
0.0457937
g(kcD) a 0.08593
*
I
]
89
0.
g(kc,D)
Q*3
3.
0.729
I
[0.1854937 x 0.234567 + 1 .
[1.0435107 - 1.0211
.
(2)
g(k,c,D) - 0.08650
(3)
g(kc,D) = 0.0519
1.0012072928923
= 141.63865 [1.0012073
]
V
1.0006035)
g(k,c,D) a 0.08552
(4)
g(k~c*D) n
0,729
[90912
0.0253289 [1.005941324 ..
n
0729
I
[1.0059413 - 1.0029638]
g(kc,D) - 0.08569
k = 0.70
(1)
g(kpD)
0.43
[1.047662816
0.057937
1.023551
g(kcD) a 0.18061
(2)
(3)
(4)
0.343
g(k,c,D)
;
g(k,c,D)
U
0.18642
g(kocD)
U
0.,343
g(kcD)
U
0.17873
g(k,c,D)
U
g(kcD)
U
0
0.17960
[1.19306481
.
1.092251
[1.005357 - 1.0026753
[1.0263627 - 1.0131
-
-
I
-4
90
k n 0.50
g(kcD) =
0.1
[1.1373811
0.0457937
-
(1)
1.0665 " - I
g(kocD) = 0.19348
-
0.*125
g(kcD) m 0.15
[ 1.556481
1.2476 - - I
10154407
1.0077 - - I
-
(2)
g(k,cD) n 0.20815
(3)
g(k,c,D) n 0.0051 69
g(k,c,D) n 0.18799
(4)
0.,125
g(kcD)
g(kcpD)
-
0.19092
[1.07 9867- 1.03730 --
I
91
BTBLI OGRA.PHY
American Institute of Steel Construction,
Steel Construction Manual,
Fifth Edition, (1947) New York, N. Y.
Bleich, Friedrich,
The Buckling Strength of Metal Structures,
Critical Survey,
Contract NObs-45424,
A
David Taylor Model
Basin, Bureau of Ships, Navy Department, with Frankland and
Lienhard.
Dinnik, A. N.,
(1949)
Design of Columns of Varying Cross Section,
Trans.
actions A.S.M.E., Applied Mechanics Division, Volume 54
(1932) pp. 165 - 171.
Formed Steel Tube Institute,
Handbook of Welded Steel Tubing.
1621 Euclid Avenue, Cleveland, Ohio. (January 15, 1941.)
Summerill Tubing Company,
Bridgeport, Conn.
Timoshenko, S.
,
Summerill Aircraft Tubing Data,
(March 8, 1943.)
Theory of Elastic Stability,
Company, Inc., New York, N. Y.
(1936)
McGraw-Hill Book