(3b
no.
OreOfl
Cction
The Use of Fourier Series in the
Solution of Beam-Column
Problems
By
B. F. RUFFNER
Professor of Aeronautical Engineering
Bulletin Series
August 1945
Engineering Experiment Station
Oregon State System of Higher Education
Oregon State College
THE Oregon State Engineering Experiment Station was
established by act of the Board of Regents of the College
on May 4, 1927. It is the purpose of the Station to serve the
state in a manner broadly outlined by the following policy:
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The Use of Fourier Series in the
Solution of Beam-Column Problems
/
By
B. F. RUFFNER
Professor of Aeronautical Engineering
Bulletin Series,
No. 21
August 1945
Engineering Experiment Station
Oregon State System of Higher Education
Oregon State College
TABLE OF CONTENTS
Page
I. Introduction
1. Introductory Statement
--------------------------------------------------------------------------------------------
5
------------------------------------------------------------------------
5
2. Summary ------------------------------------------------------------------------------------------
S
3. Acknowledgments
5
------------------------------------------------------------------------------------
II. Fundamental Considerations
------------------------------------------------------------------------
6
1. Basic Assumptions ----------------------------------------------------------------------------------
6
2. The Basic Fourier Series for the Deflection Curve ------------------------
7
III. Energy Method for Determination of Coefficients of Series ---------------1. Simple Beam-Column of Constant El with Concentrated
Loads--------------------------------------------------------------------------------------------
8
8
2. Simple Beam-Column of Constant El with Couple Applied
atEnd -------------------------------------------------------------------------------------------- 11
IV. Determination of Series Coefficients for Constant El Beam-Columns by Means of Harmonic Analysis ------------------------------------------ 13
1. Derivation of Equations ---------------------------------------------------------------------- 13
2. Example 1 ---------------------------------------------------------------------------------------- 16
V. Evaluation of Series Summations for Constant El BeamColumns Loaded with End Couples ------------------------------------------------ 18
1. Definition of Functions Pd, F, and
2. Computation of Functions Pd, P and
3. The Functions I"a, P's, and
PM ----------------------------------------------
18
M .......................................... 19
P'M ------------------------------------------------------------
20
VI. Principle of Superposition ---------------------------------------------------------------------------- 20
1. Statement of Principle -------------------------------------------------------------------------- 20
2. Use of Principle in Solution of Example 1 -------------------------------- 20
3. Example 2 -------------------------------------------------------------------------------------- 21
4. Use in Statically Indeterminate Structures -------------------------------------- 22
5. Example 3
-------------------------------------------------------------------------------------- 22
VII. Use of Series in the Solution of Problems Involving BeamColumns of Varying El ---------------------------------------------------------------------- 26
1. Theory of Method .................................................................................. 26
2. Energy Method for the Determination of the Approximate
CriticalLoad -------------------------------------------------------------------------------- 28
3. Example 4 -------------------------------------------------------------------------------------------- 29
4. Suggestions on Use of Method ---------------------------------------------------- 37
VIII. Conclusions
IX. References
37
----------------------------------------------------------------------------------------------
37
X. Appendices .......................................................................................................... 37
1. Appendix A, Table of Function rM and P'M .................................... 37
2. Appendix B, Table of Function P and P' ...................................... 37
3. Appendix C, Table of Function rd and P'd ...................................... 37
....:-
-y
.;--
i_..
-
.
'eac
ILLUSTRATIONS
Page
Figure 1. Simple Beam-Column with Concentrated Load ----------------------------------
9
Figure 2. Simple Beam-Column Loaded with End Couples ----------------------------- 12
Figure 3. Simple Beam-Column, General Case of Any Lateral Loading
.
14
Figure 4. Diagrams for Example 1 ------------------------------------------------------------------ 15
Figure 5. Diagrams for Example 3 ------------------------------------------------------------------ 22
Figure 6. Deflection Curves and Sketch of Beam-Column for Example 4 ... 30
Figure 7. Plot of
¶z\2
'2/
I\
sin
ii
for Example 4 -------------------------------------------------- 34
Figure 8. Moment Diagram for Example 4 ---------------------------------------------------- 36
TABLES
Table 1. Computations for Solution of Example 1 by Harmonic Analysis
17
Table 2. Computations for Solution of Example 1 by Use of Principle
ofSuperposition ------------------------------------------------------------------------------------------ 21
Table 3. Computations for Solutiqn of Example 2 ---------------------------------------- 23
Table 4. Computations for Solution of Example 3 ---------------------------------------------- 25
Table 5. Computations for Solution of Example 4 ---------------------------------------- 33
Appendix A. Table of Functions
Appendix B. Table of Functions
1'M
F8
and
and
Appendix C. Table of Functions F4 and
4
'M ------------------------------------------------------
F'8
F'4
38
46
54
The Use of Fourier Series in the
Solution of Beam-Column Problems
By
B. F. RUFFNER
Professor of Aeronautical Engineering
I. INTRODUCTION
1. Introductory Statement. In many engineering structures it is desirable to design members that are simultaneously subj ected to beam and column
loads. These are known as beam-columns. If the column loads are small they
will have little effect on the bending stresses. If, however, relatively large
column loads are present the bending moments in the members may be substantially greater than the bending moments produced by lateral loads only.
The analysis of this latter type of structure becomes laborious for many types
of loading. In this bulletin methods for the use of Fourier series for representing the deflection curve are discussed. It is believed that in many applications these methods are simpler, quicker, and more direct than commonly used
methods.
It is assumed that the reader is familiar with elementary beam theory
commonly discussed in standard texts on strength of materials.
2. Summary. In this bulletin are presented methods for the solution of
beam-column problems by use of the trigonometric series:
y = a, sin- +
3rx
2crz
02
sin
+ a3 sin
.
.
.
a,, sin
for representing the equation of the elastic curve. The coefficients a,, a2 .
.
.
a,,,
are constants in any particular problem. Two methods for the determination
of these coefficients are given for problems involving beam-columns of constant
El, where E is the modulus of elasticity and I the moment of inertia. For the
solution of problems of beam-columns with varying El a method of successive
approximations is given.
Tabulated values of deflection, slope, and moment functions are given for
beam-columns loaded with end couples. Use of these in the solution of beamcolumn problems with statically indeterminate reactions is discussed. Emphasis
in the bulletin is on applications of the methods rather than on development of
rigorous theorems regarding series convergence, etc.
3. Acknowledgments. Necessary financial assistance in preparing and
publishing this bulletin was provided by the Engineering Experiment Station.
The work was carried on under the general supervision of S. H. Graf, director
of the Experiment Station, who edited this report and prepared the material for
The author thanks W. E. Milne, head of the Department of
Mathematics, for his suggestions on methods of computing the values of
F, T,, etc., found in the tables. Calculations of deflection, slope and moment
functions, drawing of figures, and typing of manuscript were done by Eloise
publication.
Hout, research aide.
6
ENGINEERING EXPERIMENT STATION BULLETIN 21
II. FUNDAMENTAL CONSIDERATIONS
1. Basic Assumptions. In this report it is assumed that the differential
equation of the elastic curve of a beam-column is given by the equation:
M
d'y
(1)
El'
dx2
where x is measured from the left support and y is measured from the unloaded
position of the beam and is positive upward. The bending moment M is taken
as positive if the upper fibers of the beam are in compression. The moment of
inertia of the cross section of the beam about its centroidal axis is denoted by I.
Derivation of this equation may be found in most texts on strength of materials. Equation 1 is accurate provided the neutral axis of the beam is approximately a straight line in the unloaded position and the square of the slope of
the elastic curve in the loaded position is everywhere small compared to one.
For beam problems in which only lateral loads are acting on the structure
the bending moment is a function of x only. In that case Equation 1 may be
integrated by various methods. A useful method of accomplishing this is by
the use of a Fourier series for representing the deflection curve (1).
When axial compressive loads are acting in addition to lateral loads the
integration of Equation 1 becomes more complex. The bending moment is then
a function of y as well as of x. If M1 is taken as the bending moment due to
lateral load only, then the total bending moment M may be written as:
M=MPy,
(2)
where P is the axial compressive load acting on the span. The term "lateral
load" is used here to include all loads, couples, etc., which produce bending
moments that are a function of x only. Equation 1 then becomes:
d2yP
dz2
El
M1
(3)
El
Equation 3 is then the differential equation of the elastic curve of a beamcolumn. The integration of this equation is the problem under consideration.
For purposes of analysis two separate types of beams are considered. The
first is the type in which the stiffness El of the beam is constant between
supports. Equation 3 may then be written in the form:
+Ayf(z),
d'y
(3a)
where A is a constant.
The second type of beam to consider is one in which the stiffness
not a constant. Then Equation 3 becomes:
El is
d'y
(3b)
dx2
(1) See Section IX for references.
7
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
Equation 3a is a linear differential equation with constant coefficients. Its
f(x) will be discussed in detail. Equation 3b is a
linear equation with variable coefficients. The use of Fourier series for the
solution of this is useful and will be discussed later.
solution for any arbitrary
2. The Basic Fourier Series for the Deflection Curve. Let the deflection curve of the beam-column be represented by a sine series of form:
2rx
y = ai sin- + a2 sin- + .
5nrx
.
.
a, sin
(4)
,
1
1
1
where the subscript ,n is used to denote the last term of the series. In the most
general case m = oz. The series may be written in the shorter form as:
nm
a,,sin,
fl7TZ
y
(4a)
m. If the coordinates for x and
y are chosen so that for a single span y = 0, at z 0 and z = 1, then series 4a
satisfies these end conditions for any values of the coefficients a. y use of
one series the entire deflection curve may be represented to any desired degree
where n takes the integral values 1, 2, 3,
.
.
.
of accuracy for the entire span under consideration, even though f(v) in Equation 3a or fi(x) and f2(x) in Equation 3b are discontinuous functions.
Differentiating Equation 4a twice with respect to x gives:
¶2
d2y
dx2
22 = m
n2a,,sin-.
12
(5)
1
n=1
Substituting the right hand sides of Equations 5 and 4a in Equation 3 gives:
¶2
n=m
P
nrn
n2asin+--asin=----,
-El
El
P
n'Tx
,
n'rz
M2
1
1
n=1
n=1
or,
22
= m
c-il
2
n1
P12 \
¶'EI
Md'
asin------.--.
¶'El
n'?Tx
(6)
1
When the coefficients a are obtained, the deflection curve may be found. If
is known, the bending moment M M1 - Py
the lateral bending moment
may then be found. The determination of M is the basic problem under consideration. It is also of practical interest to determine the critical load,
P0,., at which the column becomes elastically unstable. For long columns
P
this will determine the design loads that may be placed on the structure.
M2
8
ENGINEERING EXPERIMENT STATION BULLETIN 21
III. ENERGY METHOD FOR DETERMINATION OF
COEFFICIENTS OF SERIES
1. Simple Beam-Columns of Constant El with Concentrated Loads.
Timohenko (2) discusses a method for the determination of coefficients of
series. In Figure 1 is shown the beam under consideration. Suppose the deflection curve of the beam is given by Equation 4a. If a change in any one of
the coefficients takes place, the deflection under the load Q will change a slight
amount. Also the length of the beam will undergo a slight change that will
cause the load P to mcive.
Let the change in deflection under load Q due to a change da in any one
Then:
ely.
coefficient be denoted by
da sin
flcrc
The work W done by the load Q in moving through this small displacement
'3y is:
ncTc
WQyQda,sin----- .
.
.
.
(7)
The reduction in length A of the beam may be written, for small changes, as:
A= I
1
I - Idx.
2J\dx/
(8)
0
By differentiation of Equation 4a we find:
n = m
-=- nacos,
dy
nITx
"
dx
1
1
n=1
so Equation 8 becomes:
12
[na
A=
fl7Tx
cos
I
dx.
__J
0
Evaluating the integral gives:
¶2
A=
41
n
)
m
n2&,,.
(9)
Equation 9 gives the total horizontal displacement of the end of the beam.
If only one coefficient, say a, is varied an amount da then the horizontal displacement of the beam is:
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
13X
dX=. da,=
a,.
9
Tr2n2a.
dan.
21
The work, lW', done by the load P in moving through the horizontal displacement dX is then:
W'=PdX=
P'ff2n2ada,
(10)
.
21
The total work done by the external loads is then, from Equations 7 and 10,
Pcr2n2ada,,
Qdasin----.
21
(11)
1
V
0
I
I
I
I
I
x
I
I
I
I
I
Figure 1.
I
SIMPLE BEAM-COLUMN WITH CONCENTRATED LOAD
If the beam is in a position of static equilibrium, any small change in configuration of the system produces no total change in energy. The change in
external work must then be equal to the change in strain energy stored in the
system. Considering the strain energy to be due only to bending we may write
the strain energy U as:
U
El
dx,
2
(12)
dx2
0
or,
rM2dX
U
.
(12a)
2E1
Substituting y from Equation 4a in Equation 12 and integrating (see Reference
(1) for more detail) gives:
U
UI
r4EI
U =
41
=
10
ENGINEERING EXPERIMENT STATIOr' BULLETIN 21
The change in strain energy due to a change
8U
in any one coefficient gives:
da,,
rEI
da,, na,,da,,.
(13)
21
Equating the change in strain energy to the work done gives:
6w+sw, = U,
or
n7Tc
Pr2n2a,,da,,
Q da sin
21
1
='T4EI
,a4a,,da,,,
2P
which on solution for a,, gives:
2Q13r
irE1
1
(14)
Ln(n2/3)J
1
where
/3=.
P1
(15)
ir2EI
The deflection curve of the beam is obtained by substituting
tion 14 in Equation 4a. This gives:
a,,
from Equa-
ii = m
rEI
nlTx
w7rc
1
2Q18 'ç
n'(n2/3)
sinsin-------.
(16)
/
1
n= 1
These deflections become infinite for values of /3 = 1, 4, 9, 16, etc. The
loads P producing these values of /3 are termed the critical loads. The one of
cr'EI
, which is the smallest axial load that will
most interest is the load P =
produce buckling.
To obtain desired engineering accuracy it is usually necessary to compute
only the first few terms of the series 16. In many cases the first term will suffice. It is possible to tabulate values of the summation of the series in Equation
16 for various values of (c/I), 3, and (xli). The labor involved in determining these, however, is not believed justified. For any particular case it is relatively simple to compute the deflection curve from Equation 16. Also, an approximate method for the determination of the coefficients of the series will be
given for any lateral loading. It is believed that in most cases this approximate
method will be preferable, particularly when more than one concentrated load is
acting.
Bending moments may be determined approximately by taking the second
derivative of Equation 16 and multiplying by El. This gives:
n = fl
d2y
2Q1
1
M=EI=-------dx2
(n/3)
¶2E1
n
1
n?Tc
7T
1
1
sinsin--.----.
(17)
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
11
This series, however, converges much less rapidly than the deflection series.
More accurate values of bending moment will be obtained, if only the first
few terms of the series are computed, if Equation 2 is used instead of Equation
17. Use of Equation 2 gives a better approximation for bending moments than
is obtained for deflections. The bending moments M1 may be obtained from
principles of statics.
2. Simple Beam-Column of Constant El with Couple Applied at
End. In some structures, beam-columns are subjected to end couples. It is
desirable to consider the deflection curve of a span so loaded. In Figure 2 is
shown a beam-column loaded with a couple M1 at the left end. Let the series
of Equation 4a represent the deflection curve. The energy method may again
be used to determine the coefficients a for this loading.
The work done on the beam-column by the moment M, if any one coefficient of the series is varied is:
(18)
SW---M1801,
SO1 is the change in the angle between the tangent to the elastic curve at
the left support and the x axis due to a change da,, in any one coefficient a,,.
where
For small angles SO, = S
I dy \
\ dx /o
Since
.
n = m
7Tç
dy
na,,cos,
4-dx
fl'JTX
I
,,
then at x =0,
1
¶,
/dy\ =dx
I
n = m
na,,.
"--
n==1
The change in
/ dy \
\ dx I/-
due to a change in one coefficient is:
a
/ dy
dx
501=
=,
da,1
irnda,,
.
(19)
Substituting 50, in Equation 18 gives:
Sw= M,irnda,,
The work done by the axial load P is given by Equation 10, and the strain
energy due to bending by Equation 13. Equating the total work done to the
strair energy gives:
Pr'n'a,,da,
MiTTnda,,
21
/
lr'EIn4a,,da,
21'
ENGINEERING EXPERIMENT STATION BULLETIN 21
12
Solving for a we obtain:
a,
1
2M112r
r3EI
-1
(20)
I.
Ln(n2/3) J
The deflection curve is then:
n = m
2M11
1
Tr3EI
n(n2/3)
sin.
(21)
I
n1
Ix
4
1
,z7P
I
y
1.
-1
X
I
I
I
Figure 2.
SIMPLE BEAM-COLUMN LOADED WITH END COUPLES
Similarly the deflection curve of a beam-column loaded with a couple M2
at the left end is:
n
m
2M212
1
cr3EI
n(n2/3)
sinnT(1---),
(22)
I
or in another fonn:
N
2M212
r1EI
flI
,
(-1)'
n(n2/3)
sin.
(22a)
I
51=1
The summations in Equations 21 and 22 involve only two variables, / and
(xli). Values of these summations are given later in this report.
t
.-,..
\\*
13
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
To obtain bending moments it is again recommended that Equation 2 be
used rather than the second derivatives of Equations 21 and 22 multiplied by
Taking the second derivative of Equation 21 and multiplying by El gives:
El.
n = m
dy
2M
dx2
¶
nTx
sin-.
n
n=1
This series converges slowly and unless a large number of terms is taken the
accuracy is poor. At x/1 - 0 the series gives M 0 instead of M. If sufficient terms are taken, however, it may be shown that at an infinitesimal value
of x to the right of the left support, the summation approaches 2/7T, the M
approaches the value M1. If, however, Equation 2 is used to determine the
bending moment no difficulty arises.
IV. DETERMINATION OF SERIES COEFFICIENTS
FOR CONSTANT El BEAM-COLUMNS BY
MEANS OF HARMONIC ANALYSIS
1. Derivation of Equations. Consider the beam-column shown in Figure
3 loaded so that the bending moment M, due to lateral loads only, is any known
function of x. For the simple beam-column the moment M may be readily
computed from principles of statics. Let
Md2
In the interval x = 0 to x
I we may represent z by a sine series:
n = m
(23)
Substituting from Equation 23 in Equation 6 we obtain:
n=m
n='n
r
r
LIL ne-----Iasin¶2E1J
i
1
nrx
basin-.
1
n=1
Equating coefficients gives:
b
Fl2
n2 -
¶2E1
and using Equation 15:
(24)
/3
14
ENGINEERING EXPERIMENT STATION BULLETIN 21
LOAD DIAGRAM
1
BENDING MOMENT DUE TO
LATERAL LOADS
Figure 3.
SIMPLE BEAM-COLUMN, GENERAL CASE OF ANY LATERAL LOADING.
It is seen that once the coefficients b are determined, the coefficients a,
may be found immediately. In most problems involving several concentrated
loads it is quicker to find the coefficients b by harmonic analysis of the bending
moments M1 than to solve for values of a, in Equation 14 for various concentrated loads. It is recommended that for most problems only coefficients
In a few cases it may be necessary to solve
b1, b2, b1, b4, and b5 be determined.
for a greater number of the coefficients b. These cases will occur when very
high bending moments, M1, near the ends of the beam-column exist. Reference
(1) gives formulas for determining 11 coefficients bE or S coefficients b. For
cOnvenience the formulas for S coefficients are repeated here. Let ZI, ZI, Zz,
24, 25,
tively.
be values of z at points xli equal to 1/6,
Compute v, p, q, g, and h as follows:
22i, V2 = 222,
1/3, 1/2, 2/3,
o = 2z, v4 = 224, v1 = 225.
Tabulate as follows and find Pi, Ps, qs,
q2, g,
and
h.
and 5/6 respec-
15
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
qz
Sum 2
Difference Q2
Sum Pi
Difference q
Sum p
Difference ii
The coefficients are then:
= -1/
(vs + Pi+ - P2
1
6\
=
2
2
V3
p
12
1/
b5=
pivs
6
V3
b4 =
h
12
b=1/
1
vs+Pi--------P2
2
2
6
To illustrate the use of this method an example is included (section following).
50 LB
P,
100 LB
ISO 1.3
I
I
I
I
I
30"
II
_r
I
120" _
I
I
I
60"
900
I
-1
1
I
1
LOAD DIAGRAM
EI=15L0x 106
PQ667Fr=684OLB
I-.
z
Li
0
0
X IN INCHES
MOMENT DIAGRAM
Figure 4.
DIAGRAMS FOR EXAMPLR 1.
ENGINEERING EXPERIMENT STATION BULLETIN 21
16
2. Example 1.
A simply supported beam-column 120 inches long is
loaded by concentrated lateral loads of 50 lb at 30 inches from the left support,
100 lb at 60 inches from left support, and 150 lb at 90 inches from the left support. In addition an axial compressive load P is applied to the beam-column.
Determine the critical value of P and find the bending moments for an axial
load equal to two-thirds of the lowest critical value. The El for the beam is
constant and equal to 15 X 10 lb-in2. A diagram of the beam is shown in
Figure 4.
Solution: First compute values of M, from statics or by use of charts
in Reference (1). These are entered in Table 1. Then compute coefficients
b1, b,
b5, by the scheme given above.
Values of a,, a, . . . a5 are then
found by use of Equation 24.
.
.
.
0,
05
-0.492
-0.682
v2
04
Sum -1.946
Difference q, 0.194
Sum Pi -1.174
Difference q2
-1.168
0.587
=
1.685 = -0.573
0.384 = 0.055
12
b3 =
-(-1.174 + 1.168' =
/
6\
0.001
b4 =- -0.004 =-0.00l
12
1.168-0.587 + 1.685
6
2
2
3
3
SinceP-Psr, /3-and
-0.573
=- 1.719
0.333
0.055
a2_____z0.017
3.333
-0.001
= _______ = 0.000
8.333
-0.001
a,
0.000
15.333
-0.011
a,
2 .333
= 0.000
0.190
0.194
Sum g
ft384
Difference h -0.004
6
b2
q2
q1
s
0.190
b5 =
-0.876
-1.070
=-0.011.
=
x
z
- M12
Z==-
M
v=2z
I
0
1/0
2/6
3/6
4/6
5/6
6/0
aisin1
0
0
20
40
80
2,500
4,500
8,000
5,500
3,500
80
100
120
0
0.000
-0.240
-0.438
-0.584
-0.535
-0.341
0.000
0.000
-0.492
-0.876
-1.168
-1.070
-0.682
0.000
0.000
-0.800
-1.489
-1.719
-1.489
-0.860
0.000
2x
asin-
y
M
-Py
1
0.000
0.015
0.015
0.000
-0.015
-0.015
0.000
0.000
0
-0.845
-1.474
-1.719
-1.504
-0.875
0.000
6050
8550
10,580
12,310
10,790
15,080
18,310
16,290
6,270
9,770
0
0
0
ENGINEERING EXPERIMENT STATION BULLETIN 21
18
In this example only the first two deflection coefficients are significant. The
deflections are then given by:
y=
1.719 sin + 0.017 sin
I
2crx
I
Values of y are entered in Table 1. Bending moments are obtained by:
M=MPy.
From Equation 15 for /3
1.0,
EI
p,. = ____- = 10260 lb.
12
Therefore,
P = 0.667 P = 6840 lb.
On Figure 4 are plotted bending moments M2, - Py, and M. In fairing these
curves it should be kept in mind that the bending moments are discontinuous at
points of concentrated load.
V. EVALUATION OF SERIES SUMMATIONS FOR
CONSTANT El BEAMS LOADED WITH
END COUPLES
1. Definitions of Functions P, T, and FM. The method given above
for the determination of the deflection curves of beam-columns loaded with end
couples may be used for determination of functions. Since the series for this
loading does not converge very rapidly, however, it is necessary to use a large
number of terms to obtain a good degree of accuracy. For this reason the
exact values, to four decimal places, of the series summations have been computed. Equation 21 may be written:
M11
(25)
El
where,
2'-
1
n(n2/3)
cr
sin.
(26)
I
n= 1
Taking the derivative of Equation 21 we may write:
dy
Md
dx
El
(27)
where,
n
03
2
cos.
z/3
1
r=-.2
n7Tx
I
n= 1
(28)
19
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
Taking the second derivative of Equation 21 and multiplying by El gives:
d2y
M=EI=M1rM,
(29)
dx2
where,
n= cO
=-
r,
sin
ii
2
flz_/3
';7_
(30)
n= 1
2. Computation of Functions
rd,
F, and
following formula, applicable between n= cc
(-1Ycosn8
1+202
FM.
Bromwich (3) gives the
r.
0
a7r
cosh
aO
(31)
.
sinh aT
n2+a2
= 1
Letting,
8
=(i_i),
aiV/3,
and substituting in Equation 31 leads to the relationship:
cos------n7TX
1
I rV/3(
I 1--x
\
L
crcosl
1
I
2/3
1
.
(32)
2V/3sincrV/3
Integrating Equation 32 and solving for the constant of integration so that the
summation is zero at x/l 0 gives:
F
sini
L
sin-I.rzz
n(n2/3)
1
2/3
1
2=
_I
x
TV/3( 1--
_(if)
sin iT
I
Differentiating Equation 32 gives:
n = cc
fl
L 2!3
2=
1
fllTX
1
sin[V(i -)1
2sinT\//3
(34)
20
ENGINEERING EXPERIMENT STATION BULLETIN 21
From Equations 33 and 26:
x
11/
sini cr\//3 1--_
L
(35)
sinrV
From Equations 32 and 28:
I
1
+
= -
cosL
x
'rV(" l__)1
1/
(36)
TV,8sin'?TV/3
From Equations 34 and 30:
(l)1
[
(37)
sin 7rV/3
Equations 35, 36, and 37 were used to compute values of Pd, I', and FM for
values of x/1 from 0.00 to 1.00, and for values of /3 from 0.0 to 4.0. These
are tabulated in Appendixes A, B, C. These tables are most useful in solving
problems in which beam-columns occur as members of statically indeterminate
structures.
F's,
The corresponding values for end
3. The Functions
M.
couples acting at the right end of a beam may be found by the following
d,
formulas:
Md2
dy
From Appendixes A, B, C;
'd,
(38)
El
Md
_=_____IF',.
El
dx
(39)
M = M,1"1.
(40)
F's, and
F'M
may also be obtained.
VI. PRINCIPLE OF SUPERPOSITION
1. Statement of Principle. If several lateral loads are acting on a compressed bar, the resulting deflections may be obtained by superposition of the
deflections produced by each separate lateral load acting together with the axial
load P. (For proof see Reference (2).)
2. Use of Principle in Solution of Example 1. To illustrate this principle consider Example 1. We may compute the deflections for each lateral
load acting together with the axial load P. The sum of the three deflections
will then give the total for the three acting together. Equation 16 is used to
solve for the deflections produced by each lateral load acting together with the
21
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
axial load. In Table 2 are shown computations for the deflection at the mid-
point of the beam. The first three terms of the series of Equation 16 are used
1.717 inches as
for each load. The resultant deflection at the midspan is
1.719 inches obtained previously by the method of harmonic
compared to
analysis of the bending moment curve. If end couples are acting together
Table 2.
COMPUTATIONS FOR SOLUTION OF EXAMPLE 1 BY USE OF PRINCIPLE OF SUPERPOSITION
x
1
n
2
3
4
lUTZ
flS7C
1
1
2l
3
ri-EI
1
6
5
1
fl7TC
----------
sin
sin
2
= -, ------ = 2.37 X 10_I
0.50,
1
n2(si2-
8)
sin
lUTZ
2Q15
A
sr4EI
sin
n(n2-/9)
A
X Col. (5)
y
2QP
c
= 50.0 Ib, - = 0.25, - = 0.1185
¶4E1
1.000
0.000
1
2
-1.000
3
0.707
1.000
0.707
QI
1.000
0.000
1
2
-1.000
3
1.000
0.000
-1.000
0.252
0.000
2.121
0.075
3.000
0.075
0.013
-0.001
-0.009
= 100 Ib,
c
= 0.50,
2Q11
cEI
= 0.2370
0.712
0.000
0.003
3.000
0.000
0.013
3.000
0.075
0.013
-0.715
150 lb.
= 0.75,
2Q15
0.3555
ST5EI
1.000
1
0.000
2
-1.000
3
0.707
-1.000
0.707
3.000
0.075
0.013
2.121
-0.075
-0.009
0.754
0.000
---0.003
-0.751
-1.717
-
with lateral loads it is suggested that the values given in Appendixes A, B,
and C be used for computation of the effect of the end couples and that methods
of harmonic analysis be used to evaluate the effect of the other lateral loads.
By superposition the effect of the combined loading may then be computed. For
instance consider the following example (next section).
M1
3. Example 2. The beam of Example 1 is loaded with an end couple of
1,000 lb-in. in addition to the lateral loads and axial load of 6,840 lb.
Determine the deflections at x 0, 20, 40, 60, 80, 100, 120 inches.
Solution: The deflections and moments due to the lateral loads of 50, 100,
and 150 lb acting with the axial load of 6,840 lb have been computed previously.
These are given in Table 1. To these values must be added the deflections and
moments due to the end couple M1 acting with the axial load of 6,840 lb. By
interpolation in tables of Appendixes A and C, using P
2/3, the values of
22
ENGINEERING EXPERIMENT STATION BULLETIN 21
Bending moments MM1 and deM and Pa, given in Table 3 were obtained.
were computed by use of Equations 25 and 29. The total bendflections y
ing moments and deflections given in the table are the sum of these due to the
concentrated lateral loads and the end couple.
4. Use in Indeterminate Structures.
The tabulated values of
Pa, I',,
and T'M are also useful in solving for statically indeterminate reactions. This
application is illustrated in Example 3.
5. Example 3. A beam is loaded as shown in Figure 5. The El of the
beam is 15 >< 101 and is constant over the span. If the axial load P = 6,840 lb
find the bending moment diagram for the beam.
Compute the value of the
buckling load.
LB
tOO LB
50 LB
LOAD DIAGRAM
5OLB
tOOLS
150L8
MI
P9
I
I-
x
I
LOAD DIAGRAM WITH END COUPLE M REPLAC
ING ROTATIONAL RESTRAINT AT LEFT END
X IN INCHES
'MOMENT DIAGRAM
Figure 5.
DIAGRAMS FOR EXAMPLE 3.
Table 3. COMPUTATIONS FOR SOLUTION OF EXAMPLE 2
From
Appesi-
(lix A
From
Appendix
Due to
End Couple M1
x
-
X
F,1
L
0
1/8
2/6
3/6
4/6
5/6
1.0
0
20
40
60
80
100
120
1.0000
1.5456
1.8149
1.7604
1.3304
0.8307
0.0000
0.0000
-0.1082
-0.1744
-0.1914
-0.1595
-0.0901
0.0000
I
M
y
_________
1,000
1,546
1,815
1,760
1,330
S31
0
0.000
i
-0.104
-0.167
-0.184
-.0.153
-0.086
0.000
From Table 1
M
y
0
8,550
15,080
18,310
16,290
9,770
0
___
0000
-0.845
-1.474
-1.719
-1.504
-0.875
0.000
Total
M
y
1,000
10,096
16,895
20,070
17,620
10,601
0
____
0.000
-0.949
-1.641
-1.903
-1.657
-0.961.
0.000
24
ENGINEERING EXPERIMENT STATION BULLETIN 21
Solution: To use the principle of superposition consider the beam as a
simple beam loaded with an unknown end couple M5. The couple M, must
then be such that the slope of the deflection curve at x = 0 is zero.
The equation of the slope of the elastic curve produced by the lateral loads
of 50, 100, and 150 lb is obtained by differentiation of the deflection curve obtained for Example 1 since the beam without the end couple is the same as for
that example. Therefore since,
11=5
dy
r
dx
1
fl'7T
nacos,
n= 1
then at x = 0,
11=5
/dy\
ST
)
I
\dxJ=.
Substituting values of
a1
.
ST
1 719 +
dx
(dy
/ =.
/dy\
(
I
\ dx
/
[
=-
nan.
(41)
1
n= 1
.
.
for Example 1 in Equation 41 gives:
a5
2(0.055)
+
3.333
3(-0.001)
+
4(-0.001)
8.333
+
5(-0.011)
15.333
24.333
STE
= -I - 1.719 + 0.033 + 0.000-0.000-0.002
120L
=-( 1.688 I=-0.0442.
/
120\
From Appendix B, the value of F for x = 0 and /3 = 0.667 is 0.7521. The
slope at the left end due to M1 is, from Equation 27:
/dy\
=\dx/-o El
I
I
3fl
1',
=
/
120
M1 (
\/I ( - 0.7521 1 = 6.02 X
/
\15X1OV\
10-6M1.
Since the total slope of the deflection curve at the left end must be zero, then:
6.02 X
106M1
0.0442 = 0,
or,
M1
0.0442
X 10-7,34O lb-in.
6.02
The bending moments at other values of x may then be computed as in
Example 2. The end couple
M1 is
now taken as
7,340 lb-in. These are given
in the Table 4. The total bending moments are plotted in Figure 5.
The critical load may be computed in several ways. For instance, as P
approaches the buckling load for the beam-column the bending moment M1
The slope at x = 0 due to the concentrated lateral loads only
will approach
is given by Equation 41, which by use of Equation 24 may be written:
25
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
/dy\
7T'ç,
d
nb,
(42)
.
(n2!3)
I
Differentiating Equation 21 and substituting x/l = 0 gives the slope at the left
end of the beam due to the end couple M1 as:
(dy
211I
1
=------
(43)
.
n2_p
Since the slope at the left end must be zero, then from Equation 42 and 43:
dx
tr2EJ
2M,I
,
cr
1
+-
(n2__/3)
7r2EI
1
nb,,
=0,
(fl2f3)
or
nb
M1=
¶3E1
(n2/l)
212
1
(44)
.
(nf3)
As /3 approaches 1.0 only the first terms of the series in the numerator and
denominator of the fraction are significant, so for /3 = 1.0:
Table 4. COMPUTATIONS FOR SOLUTION OF EXAMPLE 3
x
0
1/6
2/6
3/6
4/6
5/6
1.0
x
0
20
40
60
80
100
120
From
Table 6
Moments Due to
Lateral Loading
(From Table 1)
Moments
Due toM1
FM
1.0000
1.5456
1.8149
1.7604
1.3304
0.8307
0.0000
- 7,110
0
11,320
13,320
12,920
8,550
15,080
18,310
16,290
9,770
9,760
- 6,100
0
0
Total Moment
Lb-in.
7,340
2,770
1,760
5,390
6,530
3,670
0
b1.
212
Therefore, the load giving
not a buckling load for the beam-column
that will make the series in the denominator
equal to zero, however, will give a value of M1
, which represents a buckling load for the beam-column. By use of Equation 28, Equation 44 may be
/3 = 1.0 is
of this problem. A value of
/3
written
nb
¶EI
(n2/3)
M1
12
Therefore, the lowest value of /3 at which r, 10 = 0 will determine the first
buckling load. An examination of the tables in Appendix B shows that this
occurs between 1
2.0 and /3
2.08. Interpolation gives F,
= 0 at /3 = 2.047.
26
ENGINEERING EXPERIMENT STATION BULLETIN 21
P!2
then the critical load for this problem is:
Since /3 =
'7T2E1
2.047r2(15 X 10')
Per
1.44 X 10'
Per
= 21,000 lb
Ans.
VII. USE OF SERIES IN THE SOLUTION OF PROBLEMS INVOLVING BEAM-COLUMNS OF
VARYING El
1. Theory of Method. When a beam-column has varying stiffness,
El,
the basic differential equation of the elastic curve has variable coefficients and
takes the form of Equation 3b. This equation may be solved for certain particular forms of fi(x). In the general case, however, this equation is usually
solved by numerical methods. These are often laborious and give no analytic
expression for the deflection curve. It is believed that the method of solution of
Equation 3b, given here, has considerable advantage over the numerical methods
commonly used (4).
The method consists of determining, by successive approximations, the
coefficients of the basic series of Equation 1. This leads to a solution for the
deflection curve to any desired degree of accuracy. When the deflection curve
has been obtained, the bending moments at any point may be found from
Equation 2.
For simply supported beam-columns having constant El it was found that
the coefficients a of the deflection series were given in terms of the coefficients
M,12
b,, of the series representing -
by:
'r2EI
Ch
- /3
(45)
,
where, for simply supported beam-columns, /3 was the ratio of the axial load to
the critical axial load, and
n = m
n'nx
IVI,12
(46)
n=i
As a first approximation it is assumed that for beams of varying El the
relation of Equation 45 still holds. The coefficients b may be determined by
M112
harmonic analysis of the actual
values, taking into account the variation
of El. Determination of /3 for use in the equation is discussed later. This procedure is equivalent to replacing the nonlinear differential equation:
+ y,
d2y
dx2
P
M1
El
El
(47)
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
27
by the linear differential equation:
+
P
M
(EI)f,
El
d2y
dx2
(48)
,, is the El of a constant El beam-column having the same critical
load as the actual beam-column of varying El. The expression
where
n=m
nT
sin
y
obtained by the solution of Equation 48 is considered to be a first approximation
to the solution of Equation 47.
A second approximation may then be obtained by writing Equation 47:
-= MzPy
El
d2y
(49)
.
dx2
Let a second approximation to the deflection curve be:
m
ii
flTX
y'=a'srn----.
(SO)
Then Equation 49 becomes:
n=m
12 MPy
n7Tx
El
(51)
)
By harmonic analysis we now determine the coefficients
b'
to satisfy the
relationship:
fl = m
12 MPy
----
sin
b'
I
r2
(52)
El
n=I
where y is taken as the first approximation from:
m
n
flfl%
y=
sin
n= 1
Using Equation 52 in Equation 51 and equating coefficients gives:
a's,
(53)
n2
and the second approximation to the deflection curve is given by Equation 50.
28
ENGINEERING EXPERIMENT STATION BULLETIN 21
This process may be repeated until the values of the coefficients obtained
from Equation 53 show negligible change from the values used in computing
the Py term in Equation 52. When no change occurs, then the deflection
curve is the best possible for a series of the finite number of terms used in the
deflection series. The method of solution can be applied regardless of the
number of terms used in the series. For most problems sufficient accuracy
will be obtained for engineering purposes if only the first five coefficients a
are computed. When the final approximation has been obtained the bending
moments may then be computed by use of Equation 2.
It should be noted here that the final solution to the problem is independent
of the first approximation. A good first approximation, however, greatly
lessens the work involved in obtaining the final solution. To obtain a good
first approximation it is necessary to find the critical load with fair accuracy so
that / in Equation 45 may be obtained.
In many problems it is also desirable, for other reasons, to know the critical
buckling load. This may influence the maximum loads that may be applied to
the structure while providing a given margin of safety.
2. Energy Method for the Determination of the Approximate Buckling Load. The axial load causing elastic buckling of a beam-column is independent of the lateral loads and depends only on the conditions of restraint of
the column, the length of the column, and the stiffness, El. To compute this
load consider a column in equilibrium in a slightly buckled condition. Assume
no lateral loads are acting. The strain energy stored in the column must then
be equal to the work done by the axial load.
To calculate approximately the strain energy and work done by the axial
load, we may assume any reasonable expression for the buckled deflection curve
that satisfies the end conditions of the column. If the deflection ctirve assumed
differs from the true buckled deflection curve, the value of the critical buckling
load will be obtained that is higher than the true buckling load by a small
amount. Timoshenko (2) illustrates this method as applied to a column fixed
at one end and free at the other. For a pin-end column a reasonable expression
for the deflection curve may be taken as:
crx
y
a, sin,
(54)
where a, is the deflection at the center.
This expression satisfies the conditions at the end of the column, i.e., y = 0
d'y
at z0 and x-1 and the bending moment EI-0 at x=0 and z1.
dz2
If the column is rigidly encastered at the two ends we may take for the buckled
deflection curve the expression:
2Tx
Y=ai(1_cos___).
Here again the end conditions are satisfied since y = 0 at x
and dy/dx =0 at z
0 and x = 1.
(55)
0 and x
F0uIUER SERIES IN BEAM-COLUMN PROBLEMS
The work, PX, done by the axial load
P0,
P,X=
2
29
is obtained from Equation 8 as:
F0,
dy
f(
- dx.
(56)
dx
0
The strain energy U due to bending in the column is obtainable by use of the
usual expression:
dx.
2E1
U
(57)
0
For pin-end columns the moment at any point is equal to P,.,y so Equation
57
becomes:
dx
(P0,)2
(58)
El
2
Equating the strain energy to the work done gives an expression for solving
for the load required to hold the column in equilibrium in thc buckled configuration. This is the buckling or critical load, P.
Example 4, below, is included here to illustrate the foregoing method for
determining the critical load and the method for obtaining the deflection curve
and bending moments for a varying El beam-column.
3. Example 4. The beam-column shown in Figure 6 has a moment of
inertia varying from I at the supports to 12 at the center of the beam. The
beam is symmetrical about the center. The moment of inertia at any section
x is given by:
/x+d \2
(59)
)
d
The distance d, Figure 6, is determined by the condition that
beam is loaded with an axial load P
0.70
12
= 10 1.
The
and a lateral load Q at
x = 2 1/3. Find the bending moments for the beam-column.
Solution: To determine the critical buckling load, assume the buckled deflection curve is given by;
2TX
= ai Sin Differentiating this we obtain;
-=
dy
a1T
dx
1
crx
cos -,
1
:
:
IX
I
d
0i012
cr
/
IC
08
[.I
+FIRST
0.4
-7/
0.2
-0.2
4=
APPROX.
SECOND APPR
THIRD APPROX.
IH
04
0.6
-0.8
x
1.
Figure 6.
DEFLECTION CURVES AND SKETCh OF BEAM-COLUMN FOR EXAMPLE 4,
30
.0
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
and from Equation
31
56:
px=
iT\2
/
Pa21'r2
(cos_ )
2r
dx,
1,'
\
and,
PcrOi7T
(60)
P2rX =
41
From Equation
57:
U=
y2dx
(Per)2
2E
I
or,
I\ sin
I
(P,)2a12
U
2E
dx.
I
0
Multiplying the expression under the integral by
of the integral by 1/12 gives:
(P)2a121
0
and the factor in front
I sin¶\2
I.
1/ dI\1J
liz /
2E12
12/1
/
(61)
Equating the work done, PX, to the strain energy U, we obtain:
Pa1T2
(Per)2ae21,,,111'Iz
(iiff )2d(ff
2E12
41
1
0
or,
¶2E12
1
(62)
Per
212
7TX1 (z)
i7 (s,d7,
32
ENGINEERING EXPERIMENT STATION BULLETIN 21
The integral in the denominator may be evaluated either by use of
In this example the factor
-\2
112\/
given in Table 5, was plotted vs x/1 in Figure 7. The
numerical or graphical integration processes.
(_iiI(sinii
),
area under this curve was determined, by use of a planimeter, to be 0.967.
Substituting this for the integral in Equation 62 gives the critical load:
5.09
El2
Pe
/2
This example was chosen so that a comparison could be made of the results
obtained with an exact solution. Timoshenko (2) gives for the critical load for
this beam-column:
5.01
El2
Per
12
It is seen that the approximate method gives a slightly high value (1.6%) for
the critical load.
Solution for deflection curve and moments: To determine the deflection
curve of the beam a five term series for Equation 46 was used. The coefficients
b were determined in the same manner as for the constant El beam previously
discussed. The first approximation for the coefficients a in the series for the
deflection curve are obtained from Equation 45. In Table 5 the computations
are shown. For purposes of computation let:
Q13
sr2EJ2
Then using values of v in Table 5 the coefficients b,, are found by use of formulas given for five term series as:
b1 zz0.334q'
0.119q'
b2
b,
From Equation 45 using /3
=
=
0.155 q'
0.003 q'
0.013 q'
0.70 we obtain:
a1
a2
=
a3
a4 =
a5
- 1.111 q'
0.036 q'
0.019 q'
0.000
0.001 q'
These were used in the equation:
y=
sin
and values of y/q' were computed and entered in Table 5. The coefficients b'
(Equation 52) were then computed from values of v' listed in the table.
'
( JfJ2
-X
-
1
1
12
1/6
2/6
3/6
4/6
5/6
0
3.79
1.81
1.00
1.81
3.79
1.0
10.00
o
12 /
x \2
-I sin-I
1\
1/
0.000
0.948
1.358
1.000
1.358
0.948
0.000
M
2E1 J
J 2M12
7r2E1
QI
0.0000
0.0556
0.1111
0.1667
0.2222
0.1111
0.0000
-0.000
-0.211
-0.201
-0.167
-0.402
-0.421
-0.422
-0.402
-0.333
-0.804
-0.842
0.000
0.000
0.000
2.
-:
O.(
05
00
-1.252
-2.504
0.000
0.000
V.22425
45.252522
3.22244)
25.2244)
0.22424
-0.640
0.000
-0.647
0.331
0.000
0.343
0.000
0.346
0.000
0.000
I
I
-1.
,J.
C
H
ü
'0
0.2
0.4
0.6
0.8
x
t
ITZ
12
Figure 7.
PLOT OF
34
sin
2
FOR EXAMPLE 4.
1.0
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
35
These are:
b'1 = - 1.092 q'
b2=
0.154q
b', =-0.549 q'
b'4 =
0.016 q'
b',. =
0.019 q'
From Equation 53 the coefficients a',. were found to be:
a'1-1.092q'
a,.
0.038q
a'5 =
a'4 =
a', =
0.061 q'
0.001 q'
0.001 q'
The improved expression for the deflection curve is:
y' = 'a',, sin y
Computed values of - are also given in Table
5.
q
To show the convergence of the coefficients determined by this method, another approximation was made using the improved values of y' in determining
new coefficients b"5 in the series:
5
flux
b",.
Coefficients
a",
sin
1'
M, Py'
u'
El
=
were then computed from:
a",
b",.
a2
These are:
a", =
a"=
a",.
a4
=
1.095 q'
O.039q'
0.065 q'
0.001
q'
a",. = 0.001 q'
These indicate that the second approximation was sufficiently accurate for the
problem under consideration. A more exact solution could be obtained if a
series with a greater number of terms were used. It is the opinion of the
author, however, that for almost all practical stress analysis problems a five
term series is sufficient.
In Table 5 the total bending moments:
M
M, - Py
= M1
M1
(first approximation),
Py' (second approximation),
Py" (third approximation),
'WA.
o..
uI.up1M.i..
iuuurum
UVANNIUIU
IUUUNI
I-
z
LJ'
0
. ri
-
1]
JNUURUU
02
0.4
0.6
1.
Figure 8.
MOMENT DIAGRAM FOR EXAMPLE 4.
0.8
1.0
::'-'r
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
37
are tabulated. Use of the first approximation for M would be justified in many
design problems. In Figure 6 are plotted deflection curves obtained from the
three approximations. Bending moments arc plotted in Figure 8.
4. Suggestions on Use of Method. If a high degree of accuracy is
desired time will be saved by taking a 3 or 5 term series for the first approximation for the deflection curve. The second and further approximations may
then be made using a series having a larger number of terms.
When computing bending moments it is always better to solve for these by
adding the bending moments due to the lateral load to the bending moment due
to the axial load by use of Equation 2. Theoretically, bending moments can
also be obtained by taking the second derivative of the deflection curve and
multiplying by El. When this second method is used, however, the series for
the second derivative converges much less slowly than the series for the deflection. Consequently greater accuracy is obtained by the first method.
In some problems a good approximation will be obtained for the critical
load if the average El of a tapered beam is used to compute the buckling load.
If, however, sudden changes in stiffness occur, it is well to check the buckling
load by the method given above.
VIII. CONCLUSIONS
1. The methods of harmonic analysis of the bending moments due to
lateral loads provide means of computation of deflections and bending moments
for beam-columns of constant and varying El. The degree of accuracy obtainable by these methods depends on the number of terms of the series used. For
most practical problems it is recommended that a five term series be used.
2. A major advantage of the method is that a single expression for the deflection curve is obtained for the entire span regardless of the manner of lateral
loading. Once the bending moments due only to lateral loads have been cornputed, the method is as simple to apply to beam-columns with irrcgulaly vary-
ing distributed loads, or with several concentrated loads, as it is for the most
elementary loading conditions. Unless redundancies exist these bending moments due to lateral loads are easily obtainable from principles of statics.
3. The tabulated values of F, F, rM, F'd, r,, and FM greatly reduce time
in the solution of problems involving beam-columns of constant El subjected
to end moments or having rotational restraints at the ends. These tables may
be used in conjunction with the results obtained from other methods by application of the principle of superposition. This principle is particularly useful in
the solution of structures having statically indeterminate reactions.
IX. REFERENCES
(1). RUFFNER, B. F.
The Use of Fourier Series in the Solution of Beam
Problems, Bulletin No. 18, Engineering Experiment Station, Oregon
State College.
(2). TIMOSHENKO, S.
Theory of Elastic Stability, pp. 23, 27, 7, 137.
(3). BR0MwIcH, T. J. Theory of Infinite Series, p. 368.
(4). NILES, A. S., and NEWELL, J. S. Airplane Structures, Vol. II, p. 132.
X. APPENDIXES
1. Appendix A, Table of Functions FM and 1"u.
2. Appendix B, Table of Functions r, and F's.
3. Appendix C, Table of Functions F4 and F'4.
APPENDIX A
Table of Functions FM and r'M
)Ij3
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
1.0000
.9500
.9000
.8500
.8000
.7500
.7000
.6500
.6000
.5500
.5000
.4500
.4000
.3500
.3000
.2500
.2000
.1500
.1000
.0500
.0000
1.0000
.9531
.9057
.8579
.8096
.7610
.7120
.6626
.6129
.5629
.5126
.4621
.4113
.3603
.3092
.2579
.2065
.1549
.1033
.0517
.0000
1.0000
.9563
.9116
.8660
.8196
.7724
.7244
.6757
.6263
.5763
.5257
.4746
.4231
.3711
.3188
.2662
.2132
.1601
.1068
.0535
.0000
1.0000
.9595
.9177
.8745
.8299
.7842
.7373
.6893
.6403
.5903
.5395
.4878
.4354
.3824
.3289
.2748
.2203
.1655
.1105
.0553
.0000
1.0000
.9629
.9240
.8832
.8407
.7964
.7507
.7034
.6548
.6049
.5538
.5016
.4483
.3943
.3394
.2839
.2278
.1712
.1143
.0572
.0000
1.0000
.9664
.9305
.8922
1.0000
.9700
.9372
.9016
.8633
.8225
.7792
.7336
.6859
.6361
.5844
.5310
.4760
.4197
.3620
.3034
.2438
.1835
.1226
.0614
.0000
1.0000
.9738
.9442
.9113
.8753
.8363
.7943
.7497
.7024
.6527
.6008
.5468
.4909
.4333
.3742
.3139
.2524
.1901
.1271
.0636
.0000
1.0000
.9776
.9514
.9214
.8878
.8507
.8102
.7665
.7198
.6702
.6180
.5634
.5066
.4477
.3871
.3249
.2615
.1970
.1318
.0660
.0000
1.0000
.9816
.9589
.9319
.9008
.8657
.8267
.7841
.7380
.6886
.6361
.5808
.5230
.4628
.4006
.3366
.2711
.2044
.1368
.0685
.0000
1.0000
.9858
.9667
.9429
.9144
.8813
.8440
.80!5
.7570
.707S
.6551
.5992
.5403
.4787
.4148
.3488
.2812
.2121
.1420
.0712
.0000
1.0000
.9902
.9748
.9543
.9285
.8978
.8621
.8218
.7770
.7280
.6751
.6185
.5585
.4955
.4298
.3618
.2918
.2203
.1475
.0740
.0000
1.0000
.9946
.9833
.9662
.9434
.9150
.8812
.8421
.7981
.7493
.6962
.6389
.5778
.5133
.4457
.3755
.3031
.2289
.1534
.0769
.0000
/for r'M
for FM
0.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
.8518
.8092
.7647
.7182
.6700
.6201
.5687
.5159
.4619
.4067
.3505
.2934
.2356
.1772
.1184
.0593
.0000
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
APPENDIX A (Continued)
Table of Functions rM and r',
0.26
0.28
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
0.50
1.0000
.9993
.9922
.9787
.9589
.9330
.9011
1.0000
1.0042
1.0014
.9917
.9752
.9520
.9221
.8859
.8436
.7955
.7418
.6831
.6196
.5519
.4803
.4054
.3278
.2478
.1662
.0834
.0000
1.0000
1.0093
1.0111
1.0054
.9923
.9719
.9443
.9096
.8683
.8205
.7667
.7072
.6424
.5729
.4992
.4218
.3412
.2581
.1732
.0869
.0000
1.000
1.0000
1.0202
1.0319
1.0350
1.0293
1.0151
.9921
.9612
.9220
.8752
.8209
.7598
.6924
.6191
.5406
.4576
.3708
.2809
.1886
.0947
.0000
1.0000
1.0262
1.0432
1.0510
1.0494
1.0385
1.0184
.9893
.9514
.9050
.8507
.7887
.7198
.6445
.5634
.4774
.3871
.2933
.1970
.0990
.0000
1.0000
1.0324
1.0550
1.0678
1.0706
1.0634
1.0462
1.0192
.9826
.9368
.8823
.8195
.7490
.6715
.5877
.4984
.4045
.3067
.2061
.1035
.0000
1.0000
1.0389
1.0678
1.0857
1.0931
1.0897
1.0756
1.0509
1.0159
.9708
.9161
.8524
.7803
.7005
.6138
.5210
.4231
.3210
.2158
.1084
.0000
1.0000
1.0458
1.0808
1.1046
1.1170
1.1178
1.1071
1.0848
1.0514
1.0070
.9522
.8876
.8138
.7315
.6417
.5452
.4431
.3364
.2262
.1137
.0000
1.0000
1.0532
1.0949
1.1248
1.1425
1.1478
1.1406
1.1210
1.0894
1.0458
.9910
.9254
.8497
.7649
.6717
.5713
.4646
.3530
.2375
.1194
.0000
1.0000
1.0610
1.1099
1.1463
1.1697
1.1798
1.1765
1.1599
1.1301
1.0876
1.0326
.9660
.8885
.8008
.7041
.5994
.4879
.3708
.2496
.1255
.0000
1.0000
1.0693
1.1260
1.1893
1.1988
1.2141
1.2151
1.0000
1.0782
1.1431
1.1940
1.2301
1.2511
1.2566
1.2467
1.2214
1.1810
1.1261
1.0573
.9755
.8816
.7769
.6626
.5402
.4111
.2769
.1393
.0000
for rM
0.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
for r
8634
.8202
.7718
.7184
.6603
.5981
.5320
.4625
.3900
.3151
.2881
.1596
.0801
.0000
1.0146
1.0213
1.0198
1.0103
.9929
.9676
.9347
.8944
.8470
.7930
.7327
.6666
.5953
.5193
.4391
.3555
.2691
.1806
.0907
.0000
1.2017 I
1.1740
1.1325
1.0776
1.0099
.9302
.8396
.7390
.6297
.5130
.3901
.2627
.1321
.0000
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.15
.30
.25
.20
.15
.10
.05
.00
APPENDIX A (Continued)
Table of Functions I'M and I'M
x?Nj3
0.52
0.54
0.56
0.58
0.60
0.62
0.64
0.66
0.68
0.70
0.72
0.74
0.76
1.0000
1.0878
1.1616
1.2205
1.2638
1.2909
1.3015
1.2953
1.2726
1.2335
1.1787
1.1087
1.0246
.9272
.8181
.6984
.5698
.4338
.2924
.1471
.0000
1.0000
1.0980
1.1815
1.2492
1.3003
1.3340
1.3501
1.3481
1.3282
1.2907
1.2359
1.1647
1.0780
.9770
.8629
.7374
.8021
.4587
.3092
.1556
.0000
1.0000
1.1092
1.2030
1.2803
1.3398
1.3809
1.4029
1.4056
1.3889
1.3529
1.2984
1.2259
1.1384
1.0313
.9120
.7801
.6374
.4859
.3277
.1650
.0000
1.0000
1.1212
1.2264
1.3141
1.3829
1.4320
1.4607
1.4684
1.4552
1.4212
1.3668
1.2929
1.2006
1.0910
1.0000
1.1344
1.2519
1.3510
1.4301
1.4881
1.5240
1.5374
1.5281
1.4962
1.4421
1.3668
1.2712
1.1569
1.0254
.8787
.7191
.5488
.3705
.1866
.0000
1.0000
1.1488
1.2800
1.3916
1.4820
1.5497
1.5938
1.6135
1.6086
1.5791
1.5254
1.4485
1.3494
1.2297
1.0912
.9361
.7666
.5855
.3954
.1992
.0000
1.0000
1.1646
1.3109
1.4365
1.5394
1.6180
1.6712
1.6979
1.6979
1.6712
1.6180
1.5394
1.4365
1.3109
1.1646
1.0000
.8196
.6263
.4231
.2132
.0000
1.0000
1.1822
1.3452
1.4863
1.8032
1.6941
1.7574
1.7921
1.7977
1.7741
1.7216
1.8411
1.5339
1.4017
1.2468
1.0716
.8790
.6721
.4542
.2290
.0000
1.0000
1.2018
1.3838
1.5421
1.8748
1.7794
1.8542
1.8980
1.9099
1.8899
1.8381
1.7556
1.8437
1.5042
1.3398
1.1524
.9460
.7237
.4893
.2487
.0000
1.0000
1.2239
1.4267
1.6050
1.7555
1.8757
1.9638
2.0177
2.0369
2.0210
1.9703
1.8855
1.7683
1.6205
1.4448
1.2442
1.0222
.7825
.5292
.2669
.0000
1.0000
1.2490
1.4758
1.6765
1.8474
1.9856
2.0885
2.1544
2.1820
2.1710
2.1214
2.0342
1.9110
1.7538
1.5655
1.3495
1.1095
.8498
.5750
.2901
.0000
1.0000
1.2777
1.5321
1.7586
1.9530
2.1118
2.2321
2.3117
2.3492
2.3439
2.2958
2.2058
2.0757
1.9077
1.7049
1.4710
1.2104
.9278
.8280
.3169
.0000
1.0000
1.3110
1.5975
1.8540
2.0758
2.2588
2.3995
2.4952
2.5443
2.5457
2.4994
2.4083
2.2682
2.0878
1.8680
1.8133
1.3285
1.0188
.6900
.3482
.0000
forrM
0.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
'forF'M
.969
.8270
.6762
.5158
.3480
.1753
.0000
1.00
.95
.90
.85
.80
.75
.70
.85
.80
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
00
:0
Cc-,
CC
.0I
HF
c-C
¼'
CCC
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CO 0100 CO CO CO CO COO 0' CO 00 CO COO
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011000000000 a -10000-10 o
I-'
a
0-'
00a
a 00000000
0001000010100000000011000000 001000
000001000001000 010-10100001000-10'
0-00000-1100 to01000to
-o
-1 0000
a0
0-'
00
I
a
a a
a
10111
a
1-'
I
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I
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I
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aa a a
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-.a
a
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01010101010' tO 010101
I
II
111
I
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ba
0-00-'
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Ca
I
0100 to a to 001010 001-10010010010-ac
011000001010-JO to-Ia too a to
a 000
0-000000 00-10000 tOO)-' 0010 01001010-to
0010 a to a ao'co 010000000 000'a
ao
I
-i010 0-to 0000010001000000100000001
0100000100101100100 01 0' tO 001 00-0--I
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010 tot-' 00010-'-.) CII-'
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1-0011010
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010000 000-1 10 10000 -o 0000101010000010
0101to00.-Itoto000toO 0-' 10 0' 01 o 01000
100000010000001011001000100 - 000-It-'
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a
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01
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a a ooat-'to -1 01 0000001-1011101-00-'
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coo
0)-' a 01-1000000)-' a 0100 00 to 0010
0000 100001-1
01001 a a 010-100100001
too' toto a totototototo a a toeo
1-'
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000
0100 000'-' to a a -11-001-to 0C-1O tO 0001
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01
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to
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0 a toto 0.1010
I
-'tOtoa010-i-10100-I--1000olatoh-'
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tOO
0000010 1-00-' 10 01011 0010000010-100000'
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0001-' 01-'
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0010010000000 tO
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0
/
APPENDIX A (Continued)
Table of Functions rM and r'M
1.52
1.56
1.0000
.7670
.5052
.2246
1.0000
.7847
.5392
.2731
- .0644
- .3510
- .6245
- .8746
-1.0920
-1.2686
-1.3978
-1.4747
-1.4965
-1.4623
-1.3735
-1.2333
-1.0470
- .8215
- .5654
- .2881
- .0036
- .2800
- .5458
- .7906
-1.0051
1.60
1.64
1.68
1.0000
.8152
.5975
.3557
.0096
1.0000
.8287
.6231
.3918
.1443
1.72
1.76
1.80
1.84
1.88
1.92
1.0000
.8531
.6692
.4564
.2238
1.0000
.8643
.6903
.4858
.2598
.0223
1.0000
.8750
.7104
.5137
.2938
.0605
1.0000
.8853
.7297
.5404
.3261
.0968
1.0000
.8953
.7484
.5661
.3571
.1313
- .2162
- .4451
- .6544
- .8346
- .9780
-1.0780
-1.1304
-1.1328
-1.0850
- .9892
- .8496
- .6725
- .4656
- .2380
.0000
- .1754
- .4035
- .6132
- .7953
- .9414
-1.0449
-1.1011
-1.1076
-1.0639
- .9721
- .8364
- .6628
- .4593
- .2349
- .1371
- .3646
- .5752
- .7593
- .9083
-1.0153
-1.0754
-1.0858
-1.0460
- .9579
- .8256
- .6551
- .4543
- .2325
.0000
- .1007
- .3280
- .5398
- .7261
- .8781
- .9888
-1.0527
-1.0670
-1.0310
- .9463
- .8169
- .6490
- .4505
- .2307
1.96
2.00
for r'1
for rM'0.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
.0000
--1.1810
-1.3115
-1.3918
-1.4187
-1.3911
-1.3102
-1.1790
-1.0025
- .7876
- .5425
- .2765
.0000
1.0000
.8006
.5697
.3164
.0507
- .2171
- .4763
- .7168
- .9290
-1.1047
-1.2370
-1.3205
-1.3521
-1.3305
-1.2566
-1.1332
- .9652
- .7592
- .5234
- .2669
.0000
- .1605
- .4142
- .6511
- .8618
-1.0378
-. .1091
-1.2587
-1.2948
-1.2786
-1.2109
-1.0944
- .9337
- .7353
- .5073
- .2589
- .3580
- .5922
- .8023
- .9785
-1.1146
-1.2047
-1.2450
-1.2339
-1.1719
-1.0614
- .9070
- .7152
- .4939
- .2522
.0000
.0000
--1.1719
1.0000
.8412
.6469
.4252
.1855
- .0620
- .3069
- .5388
- .7480
- .9255
-1.0638
-1.1572
-1.2017
-1.1953
-1.1384
-1.0333
- .8845
- .6984
- .4827
- .2466
.0000
- .0184
- .2599
- .4901
- .6991
- .8778
-1.0186
-1.1153
-1.1637
-1.1618
-1.1096
-1.0094
- .8655
- .6842
- .4733
- .2419
.0000
.0000
.0000
1.0000
.9050
.7665
.5909
.3871
.1645
- .0660
1.0000
.9146
.7842
.6153
.4161
.1965
- .0328
- .2933
- .5065
- .6953
- .8506
- .9650
-1.0328
-1.0509
-1.0184
- .9369
- .8102
- .6444
- .4477
- .2294
.0000
- .2604
- .4753
- .6668
- .8255
- .9437
-1.0154
-1.0373
-1.0082
- .9296
- .8052
- .6413
- .4459
- .2286
.0000
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
AI'PENDIX A (Continued)
Table of Functions rM and F'M
2.08
2.16
1.0000
.9332
.8188
.6625
.4723
.2580
.0305
1.0000
.9516
.8527
.7086
.5269
.3172
.0907
- .8255
- .9437
-1.0154
-1.0373
-1.0082
- .9296
- .8052
- .6413
- .4459
- .2286
- .1985
- .4174
- .6150
- .7811
- .9073
- .9872
-1.0166
- .9940
- .9207
- .8003
- .6390
- .4451
- .2284
.0000
.0000
- .1406
- .3645
- .5690
- .7433
- .8782
- .9665
-1.0035
- .9873
- .9187
- .8013
- .6414
- .4475
- .2299
.0000
2.00
2.24
2.32
2.40
2.48
2.56
2.64
2.72
2.80
2.88
2.96
1.0000
1.0081
.9567
.8490
.6913
1.0000
1.0282
.9938
.8989
.7492
.5540
.3250
.0762
1.0000
1.0494
1.0328
.9514
.8102
.6180
.3871
.1318
1.0000
1.0720
1.0746
1.0075
.8752
.6861
.4526
.1898
1.0000
1.0964
1.1196
1.0681
.9453
.7594
.5229
.2514
- .1772
- .4198
- .6368
- .8151
- .9438
-1.0150
-1.0244
- .1318
- .3871
- .6180
1.0000
1.1852
1.2844
1.2904
1.2027
1.0276
.7780
.4719
.1316
.0000
- .2896
- .5739
- .8188
-1.0075
-1.1270
-1.1690
-1.1308
-1.0149
- .8292
- .5866
- .3037
.0000
-.2551
.0000
- .0368
- .3226
- .5868
- .8119
- .9828
-1.0881
-1.1208
-1.0787
- .9646
- .7861
- .5552
- .2872
.0000
1.0000
1.1524
1.2233
1.2079
1.1071
.9281
.6836
.3908
.0703
- .9514
-1.0328
-1.0494
-1.0000
- .8878
- .7198
- .5066
- .2615
- .0853
- .3549
- .6015
- .8091
- .9643
-1.0570
-1.0813
-1.0355
- .9226
- .7499
- .5287
- .2732
1.0000
1.1230
1.1689
1.1344
1.0221
.8395
.5993
.3179
.0147
- .2183
- .5524
- .8464
-1.0789
-1.2331
-1.2978
-1.2682
-1.1467
- .9418
- .6686
- .3469
.0000
forrM
forrM
0.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
.9146
.7842
.6153
.4161
.1965
- .0328
- .2604
- .4753
-.6668
1.0000
.9701
.8867
.7546
.5810
.3754
.1492
.0853
- .3150
-
.5274
.7108
.8552
.9524
.9973
.9873
.9229
.8078
.6483
.4531
.2330
.0000
1.0000
.9888
.9212
.8012
.6355
.4336
.2070
- .0313
-
.4928
.2653
.0222
.2679
.4893
.6827
.8373
.9442
.9972
.9935
.9331
.8196
.6595
.4617
.2376
- .2223
- .4536
- .6583
- .8241
- .9414
-1.0032
-1.0059
- .9498
- .8368
.0000
.0000
- 6749
- .4733
- .2439
- 9715
-
.8594
.6950
.4882
.2518
.0000
.8102
- .5625
- .8301
-1.0392
-1.1748
-1.2274
-1.1933
-1.0749
- .8806
- .6241
- .3234
.0000
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
.
..-
to
to
CO
a
CC
0)
-i
00
03
0)
00
C
03
to
Cli
03
-t
0
.
.'.1c0G,l1-].o
.10---(o
000000I100I1001OC)I000iOOlOOlOOiO
eet-'l.-0000001aacco,100-I-I0000C000
1-4
g888888S88888888888
1-'
1-' 1-'
0-'
Cli
Cli
I
I
I
I
1100011
I
I
I
I
-'0 cia 00
(0 00000 1DeC11 (0000 01 0010
00000000 00 tO 0000001-' 00000.0 0000
0000100 00.0000004000.0.0cc a 00000 a 0
0)00010-40 a GO CO 11-400000 -1070 000110
10111-4
1-'
COO 00 0000000100 01 oo a 0-1-10000-000000
0.00000 CO a 0000I100tO 00 000010000
0000ot00000000-'a00000001oiatoo
l.'.001
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I
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0.0
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a 00010
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t
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011
I
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0010010010010010010010010010010
O3010100
I
tOO 01 00 1-' a 00110000100-100000 0 0-ito
0101001000010101010-3004010101cc ci 0
.0.
0000000000001010101010000010100010
tO 00 03
tO 001
00Q00000000001O
I
0-100010-10010001.0.0 a 0000000000
00010 OcC01000001000-itO 0)C-& .0-1-.] 0000)
00 CO a a 000)001
00000090100100000400-.1000
01 00 01-' a 04.0
ao
0.010
010001010000.000-100000010
C-'0 tO 110 00 tO 001 -i 000000 0)01-' C-' 0) 001
01000000 a 0-' 01000..] to to a 0)0) a 00.0001
01
0-' 0' C-'
01010000101
01)0 001-'
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tO 01-00-1010.01-'
40 00-Ia 001-' a t000-I000000.0t.000 COCO
0000 10010014-I a 0000000-1.300000010
1--''1-'1--'1--'C-'1--'-'
0004t-'Oocitoop0oo-ItocO000Oo-'00010000
a to to CoO -&
00 -1 0-1
0.00101000 01CC-to a-I
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00 000
0.0-1
0 1-.t.000a0000-ia
01-I
0000.4Otot000.-'ont-'-I0000-'tot000iO
00101-10100-10001040 a a 00001.0cc
at
0001000.0010.000000101000 a a
000l00000i0001000-1-I100 a a 040110-400
t001tO010 0' 0'
0-'
00100-1010 a-I 00000 0403 a 00100100
0000100000t000000.0010t001000000
0'00 01-1001 000000 00010100 a 0-lao
000001-1000100001-00.0010 0000,-'a 01-c
I
.00000000 Cli Cli 00 3000000010 C)1 -I 0I
o
a
woo 1-4-lao 001000000000000 00000
a aGO 001-10001
00000t01-'00001010000
Coo 001-'-I000001 a t-oo o'0-I
b
to tO 00 00 tO 0.0 1-'
I-' tO tO 0.00003 t'O 0-'
1-' 1-'
0.001010100.0000-' I-'
0.00 a o
00 00001101 0Cli-ICJiOOOOCO-Iao
COO a .1-I0O CO
t00t001-'a0001° CC000000
000 o-1000000000o a 000000001010000
00)000
o1-'110000000000oi0000-la
GO
0000a001.0000.oI-.
0010000C-'t000-10000oo0100i-'tOt000-I000JiO
0
00000 003 toOl' 00 10 00000 000-i
0000000-100 00 00 000 a 0400010100
b .401 0-40) t3 007100-3 o a a 0110)-'otoo
0.00-'
1100110041
I
I
0;C,*4L,000...--
010
0
0z
a
a
03
00
03
00
tO
04
0
00
CO
03
0'
CO
.0
0
1
/z
/
/
ono.._,.'..
APPENDIX B
Table of Functions r and -r'
0.00
for r
0.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
N
-for -F's
-0.3333
- .2846
- .2383
- .1946
- .1533
- .1146
- .0783
- .0446
- .0133
.0154
.0417
.0654
.0867
.1054
.1217
.1354
.1467
.1554
.1617
.1684
.1667
-0.3378
- .2890
- .2424
- .1984
- .1567
- .1174
- .0806
- .0463
- .0144
.0151
.0421
.0663
.0882
.1075
.1242
.1383
.1499
.1590
.1654
.1693
.1707
-0.3425
- .2936
- .2468
- .2024
- .1603
- .1205
- .0830
.- .0481
- .0155
.0146
.0421
.0672
.0896
.1095
.1267
.1413
.1533
.1626
., .1693
.1733
.1747
-0.3473
- .2983
- .2514
- .2065
- .1639
- .1236
- .0855
- .0499
- .0166
.0141
.0424
.0681
.0912
.1116
.1294
.1445
.1569
.1665
.1734
.1776
.1789
.1677
.1268
.0881
.0518
.0178
-0.3576
- .3084
- .2610
- .2154
- .1718
- .1302
- .0909
- .0538
- .0191
-0.3630
- .3138
- .2661
- .2201
- .1759
- .1338
- .0937
- .0559
-.0204
-0.3687
- .3194
- .2714
- .2250
- .1804
- .1376
- .0968
- .0582
- .0219
.0137
.0427
.0691
.0928
.1139
.1322
.1478
.1606
.1706
.1777
.1820
.1834
.0132
.0429
.0700
.0945
.1162
.1351
.1512
.1644
.1748
.1821
.1866
.1881
.0126
.0432
.0710
.0962
.1186
.1382
.1548
.1685
.1792
.1868
.1914
.1930
.0120
.0434
.0721
.0980
.1212
.1413
.1586
.1727
.1838
.1917
.1965
.1981
-0.3523
- .3032
- .2560
-.2108
-
.
-0.3747
- .3252
- .2770
- .2302
- .1849
- .1414
- .0999
- .0605
- .0233
.0114
-0.3809
- .3314
- .2828
- .2355
- .1897
- .1455
- .1032
- .0629
- .0249
.0108
.0437
.0732
.1000
.1238
.1447
.1625
.1772
.1886
.1969
.2018
.2035
.0439
.0744
.1020
.1266
.1482
.1667
.1819
.1938
.2023
.2074
.2091
-0.3875
- .3378
- .2890
- .2412
- .1948
- .1499
- .1067
- .0655
- .0265
.0101
.0442
.0756
.1041
.1296
.1519
.1710
.1868
.1991
.2079
.2133
.2151
-0.3943
- .3446
- .2954
- .2472
- .2001
- .1544
- .1104
- .0683
- .0283
.0094
.0445
.0768
.1063
.1326
.1558
.1756
.1919
.2047
.2139
.2195
.2213
-0.4015
- .3517
- .3022
- .2534
- .2057
- .1592
- .1143
.0711
- .0301
.0086
.0447
.0781
.1086
.1358
.1598
.1804
.1974
.2107
.2202
.2260
.2279
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
APPENDIX B (Continued)
Table of Functions 1'
0.26
forI
0.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
0.28
0.30
0.32
0.34
0.38
0.36
and -F's
0.40
0.42
0.44
0.46
0.48
0.50
-..
-0.4091
- .3591
- .3093
- .2600
- .2115
- .1642
- .1183
- .0742
- .0321
.0077
.0450
.0795
.1110
.1392
.1641
.1854
.2031
.2169
.2269
.2329
.2349
-0.4171
-
.3670
-
.3168
.2669
.2177
.1695
.1227
.0774
.0342
.0068
.0453
.0809
.1135
.1428
.1686
.1908
.2091
.2235
.2339
.2401
.2422
-0.4255
- .3753
- .3247
- .2743
- .2243
- .1752
- .1272
-
.0809
- .0364
.0059
.0456
.0824
.1162
.1466
.1734
.1965
.2156
.2305
.2413
.2478
.2500
-0.4344
- .3840
- .3331
- .2820
- .2313
- .1811
- .1321
- .0845
- .0387
.0048
.0458
.0840
.1190
.1506
.1785
.2025
.2223
.2380
.2492
.2560
.2583
-0.4438
- .3933
-.3420
-
.2903
.2386
.1875
.1372
.0884
.0413
.0037
.0461
.0857
.1220
.1548
.1838
.2088
.2295
.2458
.2576
.2647
.2670
-0.4538
- .4031
- .3514
- .2990
- .2464
- .1942
- .1427
- .0925
- .0439
.0025
.0464
.0874
.1252
.1593
.1895
.2156
.2172
.2542
.2665
.2739
.2764
-0.4644
-.4136
- .3614
- .3082
-.2547
-
.2013
.1486
.0969
.0468
.0012
.0467
.0893
.1285
.1641
.1956
.2228
.2454
.2632
.2760
.2837
.2863
-0.4757
- .4247
- .3720
- .3181
- .2636
- .2090
- .1548
- .1016
- .0499
- .0002
-0.4878
- .4366
- .3834
- .3287
- .2731
- .2172
- .1615
- .1067
- .0532
- .0017
-0.5006
- .4493
- .3955
- .3400
- .2832
- .2259
- .1687
- .1121
- .0568
- .0033
-0.5145
- .4629
- .4086
- .3521
- .2941
- .2354
- .1764
- .1179
- .0606
- .0051
-0.5293
- .4775
- .4226
- .3652
- .3059
- .2455
- .1847
- .1242
- .0648
- .0071
-0.5453
- .4933
- .4377
- .3792
- .3186
- .2565
- .1937
- .1311
- .0693
- .0092
.0470
.0913
.1321
.1692
.2020
.2304
.2541
.2727
.2861
.2942
.2969
.0473
.0934
.1359
.1746
.2090
.2387
.2634
.2829
.2970
.3055
.3083
.0476
.0956
.1400
.1804
.2163
.2474
.2734
.2938
.3086
.3175
.3205
.0479
.0980
.1444
.1866
.2243
.2569
.2841
.3056
.3211
.3305
.3336
.0482
.1005
.1490
.1933
.2328
.2671
.2957
.3183
.3346
.3445
.3478
.0486
.1032
.1541
.2005
.2420
.2781
.3082
.3320
.3492
.3596
.3631
>7'x/1
for -F's
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
APJ'END!X E (Continued)
Table of Functions F and -F'.
0.50
0.52
0.54
0.56
0.58
0.60
0.62
064
0.66
0.68
0.70
0.72
0.74
for r,
0.00
.05
.10
.15
.20
.28
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
I3,77z/l
for -F
-0.5453
- .4933
- .4377
- .3792
- .3186
- .2565
- .1937
- .1311
- .0693
- .0092
.0486
.1032
.1541
.2005
.2420
.2781
.3082
.3320
.3492
.3596
.3631
-0.5626
- .5104
- .4541
- .3945
- .3323
- .2683
- .2035
- .1385
- .0742
- .0115
.0489
.1061
.1595
.2084
.2520
.2900
.3217
.3469
.3650
.3760
.3797
--0.5814
-
.5289
.4718
.4110
.3472
.2813
.2141
.1466
.0796
.0140
.0492
.1093
.1654
.2168
.2629
.3029
.3365
.3630
.3822
.3939
.3978
-0.6019
- .5491
- .4912
- .4290
- .3635
- .2954
- .2257
- .1554
- .0855
- .0168
.0495
.1127
.1718
.2261
.2747
.3171
.3526
.3807
.4010
.4134
.4175
-0.6242
- .5711
- .5124
- .4488
- .3813
- .3108
- .2384
- .1651
- .0919
- .0199
-0.6488
- .5954
- .5356
- .4705
- .4009
- .3278
- .2524
- .1758
- .0991
- .0237
.0498
.1164
.1788
.2362
.2877
.3326
.3702
.4000
.4216
.4347
.4391
.0502
.1205
.1865
.2473
.3019
.3496
.3896
.4213
.4443
.4583
.4629
-0.6759
- .6221
- .5613
- .4944
- .4225
- .3466
- .2679
- .1876
- .1070
- .0272
.0505
.1250
.1950
.2596
.3177
.3684
.4110
.4449
.4694
.4843
.4893
-0.7080
- .6518
- .5898
- .5210
- .4465
- .3675
- .2852
- .2008
- .1158
- .0315
.0509
.1299
.2044
.2732
.3351
.3893
.4349
.4711
.4973
.5133
.5186
-0.7395
- .6849
- .6216
- .5507
- .4734
- .3908
- .3044
- .2156
- .1257
- .0363
.0512
.1354
-0.7772
- .7221
- .6574
- .5841
- .5036
- .4171
- .3262
- .2322
- .1369
- .0418
.0516
.1415
-0.8199
- .7642
- .6979
- .6220
- .5378
- .4469
- .3508
- .2511
- .1496
- .0480
.0519
.1484
.2149
.2884
.3547
.4127
.4615
.5004
.5286
.5457
.5514
.2266
.3054
.3766
.4390
.4916
.5334
.5637
.5822
.5883
.2399
.3248
.4015
.4688
.5256
.5708
.6036
.6235
.6302
-0.8686
- .8123
- .7441
- .6652
- .5770
- .4810
- .3790
- .2728
- .1642
- .0552
.0523
.1563
.2551
.3469
.4300
.5029
.5645
.6136
.6492
.6709
.6782
-0.9247
- .8677
- .7974
- .7150
- .6220
- .5203
- .4115
- .2977
- .1810
- .0635
.0526
.1654
.2726
.3723
.4627
.5423
.6094
.6629
.7019
.7255
.7335
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
0
00
c-)0
'-'O
0
0
tO
00
H
-
00
CC
0
0
CO
o
C
o
CII
CO
O
0
00
0
CC
00
0
00
00
I
00000000000000000L000000000
CON
.1
ggggggssssss8sgs8s
JIllIllIll
100000 .ONO 00000000000 COOt-CO
COOt- .1000000
N 0000 00 '0 00000,-C
tO'
.-.1000000N0000000
00.-I 0000000000000000000000 N 00000
00.-CO 0.1 .-C 00000000.0000 0.1004000
I
I
I
I
I
I
I
I
I
.1 C') 00000O'0000 CO
000.-CON-CO ItO 000000000000 N 000000
N N Cot- - .00000000000000000000 N
0 N 00010010000000 COO 0100 N N CO N 000.1
CO3-t 00000000 NO 0000000000 N 00
00000000000001.0
.1 N 00 O'0000000000000000 .-C .-4 000<1
0001010
NO 00000000000000000
I
I
I
I
I
I
I
I
,-1 .000000O CO CO
000000000000100000N0,-40000
0100 N 00011000 N 00000 N N 100 N .0 N N 0001
0400001
00 00000001 CII .1 ,-4
.0.0 NOON ON CON 00.100010001 .-R0 01 00 .-0
I
I
I
00000000 ON 01O00001000 N
I
0O'0000.000.0000001 NO 00 OlIN CON O00
0000
N
0000
000.0000000,-ON
00000000
0000010 NO' 0 N 0000000.-tO' N 00000000
010110101.-t.1.0
,.00O,00101010101
III
I
-00
10 000N, ,-000.-00000000 00000 N 0104000 N
N0001 000000000 000000004 ,-INOO 0000.
.1.100000000 .-O 00100100000.-I 000 N 0000
0C.jtCj0.
0000 N 000 00100000000' NO.0000000
N
000000000000000 N 000,-.'0.-. 0
I
I
I
000000000000 tO 000 .-4 000 00 01000 N000O'
CON 01000000 N 0000000 ION 0.000000000
N 000000000 N 000 N 000.0000000000
0000,0001000 tO tO 00.1000000000000
000 00 .000000 00000 NOON 0-000 tO 00000
I
I
I
I
000000 .000100 00 .000400000,1 04 000000
00.-4L0010000L000001000000000000
I
01 01
40100O0 NON 000000000000000
.400000.0 001.-I 000000' C)O00000000,-t
I
00000.-4000LOCOCtOO'0'LO N 000.-.
0O.-,1-0C0C0C000I0I000NNC0OO000
0
OL'OOLOOIOOIOOIOO LO 00000=00000
lIIiIrlir
010000 - N 000000000000100'000.-I
000000000 N 000 00 tO) 01 01 .-0 00 .-0000000
00010
N N 00000 NON 0000004000
000O.0O0100.)C1000t00NNNN
'ii1iiriiiir
00000 N 00000 0<100,-C 0000100 N N 00000
0000 N 00000 tOO .-400001 000010000 CC .-t000
N 001 .-00000,000000000 N 00000000000
0000000010004 00 00 00 .-1 00000000 00 000 N
0001 Cl .-t 300001001000' N COON 000000
.-0O0000001000000000
N N 0000
0 = 000.-I tIC 00 00 0<000,0 00000 N 0000000
0 00 0 '1' '0 00 0 01000 00000 10 01000 100 C')
i1r'i1IIII
t 00000000 N-Cl .0 C0t CON .00100000
COCI 00 000100000000000000 N 00000
N .-0 01 0100.-I 00.400010000000 N-COON
N 00000 .1001 00000 N 00000
I
00)
Cl 0') ,-400
0,0-.
I
01
00
0
0/0
00
N
0
N
/
/
/ "0
APPENDIX B (Continued)
Table of Functions F and -F's
1.04
1.08
1.12
1.16
1.20
1.32
1.28
1.24
1.36
1.40
1.44
1.52
1.48
for -F'.
for r
4.9134
4.8990
4.7567
4.4902
4.1061
3.6144
3.0276
2.3608
1.6311
.8571
.0587
0.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
.
- .7437
-1.5296
-2.2788
-2.9721
-3.5919
-4.1221
-4.5493
-4.8625
-5.0537
-5.1180
2.3787
2.3956
2.3463
2.2321
2.0561
1.8230
1.5388
1.2113
.8490
.4617
.0596
- .3465
- .7460
-1.1281
-1.4827
-1.8004
-2.0727
-2.2924
-2.4536
-2.5521
-2.5852
1.5332
1.5606
1.5425
1.4793
1.3729
1.2261
1.0430
.8287
.5890
.3306
.0606
- .2136
- .4844
- .7443
- .9862
-1.2035
-1.3900
-1.5407
-1.6514
-1.7190
-1.7418
1.1100
1.1427
1.1402
1.1027
1.0312
.9278
.7954
.6378
.4594
.2655
.0615
- .1467
- .3533
- .5523
- .7380
- .9051
-1.0488
-1.1652
-1.2507
-1.3030
-1.3206
0.8554
.8912
.8983
.8763
.8260
.7488
.6469
.5235
.3821
.2269
.0625
- .1062
- .2743
- .4367
- .5888
- .7260
- .8442
- .9400
-1.0105
-1.0536
-1.0681
0.6856
.7236
.7371
.7255
.6894
.6297
.5483
.4477
.3309
.2015
.0635
- .0789
- .2215
- .3597
- .4895
- .6068
- .7082
- .7904
- .8509
- .8880
- .9005
0.5634
.6030
.6212
.6172
.5914
.5444
.4777
.3936
.2945
.1837
.0646
- .0501
- .1833
- .3043
- .4182
-
.5214
.6107
.6833
.7368
.7606
.7806
0.4716
.5124
.5341
.5360
.5179
.4805
.4251
.3533
.2675
.1706
.0656
- .0439
-. .1546
-
.2627
.3648
.4575
.5379
.6033
.6516
.6812
.6911
0.3999
.4417
.4662
.4726
.4607
.4309
.3842
.3221
.2468
.1607
.0668
- .0319
- .1320
- .2302
- .3232
- .4078
- .4813
-
.5412
.5855
.6127
.6218
0.3422
.3848
.4116
.4217
.4149
.3912
.3516
.2974
.2304
.1531
.0679
- .0221
- .1138
- .2041
- .2898
- .3681
- .4362
- .4917
- .5328
- .5581
- .5666
0.2947
.3380
.3667
.3800
.3774
.3588
.3251
.2774
.2173
.1470
.0691
- .0138
- .0987
- .1826
- .2625
- .3356
- .3993
- .4514
- .4900
- .5137
- .5216
0.2549
.2987
.3291
.3451
.3460
.3318
.3031
.2608
.2065
.1423
.0703
-
.0067
.0859
.1646
.2397
.3086
.3687
.4179
.4544
.4768
.4844
0.2209
.2652
.2971
.3154
.3194
.3090
.2846
.2470
.1977
.1385
.0716
-
.0005
.0750
.1492
.2203
.2857
.3429
.3897
.4245
.4459
.4531
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
APPENDIX B (Continued)
Table of Functions F and -F's
1.52
for r1
1.00
1.60
1.64
1.68
1.72
1.80
1.76
1.84
1.88
1.92
2.00
1.90
&7x/1
'for -.r',
0.2209
.2652
.2971
.3154
.3194
.3090
.2846
.2470
.1977
.1385
.0716
0.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95-
1.56
.
-
.0005
.0750
.1492
.2203
.2857
.3429
.3897
.4245
.4459
.4531
0.1915
.2363
.2695
.2898
.2966
.2895
.2688
.2353
.1902
.1354
.0729
.0051
- .0654
- .1359
- .2036
- .2661
- .3208
- .3657
- .3990
- .4196
.- .4265
0.1658
.2110
.2453
.2676
.2768
.2726
.2552
.2253
.1840
.1330
.0742
.0101
- .0570
- .1242
- .1891
- .2491
- .3017
- .3450
- .3771
- .3969
- .4036
0.1430
.1886
.2240
.2479
.2593
.2578
.2434
.2167
.1787
.1311
.0756
.0147
- .0494
- .1139
- .1764
- .2342
- .2851
- .3270
- .3581
- .3774
- .3838
0.1227
.1686
.2050
.2305
.2439
.2448
.2331
.2092
.1743
.1296
.0771
.0189
- .0426
-
.1047
.1651
.2211
.2705
.3112
.3415
.3003
.3666
0.1044
.1506
.1879
.2148
.2301
.2332
.2240
.2028
.1705
.1285
.0786
.0229
- .0363
- .0965
- .1550
- .2095
- .2576
- .2973
- .3270
- .3452
- .3514
O.O87
.1341
.172
.200
.217t
.222
.215
.1971
.1671
.1271
.0801
.026t
-
.030(
.088
.145
.1991
.2461
.285(
.3141
.332C
.3381
0.0726
.1193
.1583
.1879
.2066
.2136
.2088
.1922
.1646
.1272
.0817
.0302
- .0253
- .0821
-
.1377
.1898
.2359
.2741
.3026
.3203
.3263
0.0585
.1056
.1454
.1761
.1964
.2053
.2024
.1879
.1623
.1270
.0834
.0336
- .0203
- .0757
- .1302
- .1813
- .2267
- .2643
- .2925
- .3099
- .3158
0.0456
.0929
.1334
.1653
.1871
.1977
.1967
.1841
.1605
.1269
.0851
.0369
-
.0156
.0698
.1233
.1736
.2184
.2556
.2834
.3000
.3065
0.0335
.0811
.1223
.1553
.1785
.1908
.1915
.1808
.1590
.1272
.0869
.0401
- .0111
- .0644
- .1170
- .1666
- .2109
- .2477
- .2753
- .2924
- .2982
0.0222
.0700
.1120
.1460
.1706
.1844
.1869
.1779
.1578
.1276
.0888
.0432
- .0069
0.0115
.0596
.1022
.1374
.1632
.1786
.1827
.1754
.1569
.1282
.0908
.0464
- .0028
-
-
.0592
.1111
.1602
.2041
.2400
.2680
.2850
.2908
.0544
.1057
.1543
.1979
.2342
.2615
.2784
.2842
1.00
.95
.90
.85
.80
.75
.70
.05
.60
.55
.50
.45
.40
.35
.30
.23
.20
.15
.10
.05
.00
APPENDIX B (Continued)
Table of Functions 1
2.08
2.00
2.24
2.16
2.48
2.40
2.32
and -I"
2.72
2.64
2.56
2.88
2.80
2.96
for -Y'
for F N
0.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
0.0115
.0596
.1022
.1374
.1632
.1786
.1827
.1754
.1569
.1282
.0908
.0464
- .0028
-0.0082
-
-
.0544
.1057
.1543
.1979
.2342
.2615
.2784
.2842
.0403
.0843
.1215
.1500
.1683
.1756
.1714
.1559
.1300
.0949
.0525
.0050
.0453
.0958
.1439
.1871
.2233
.2505
.2674
.2731
-0.0262
.0228
.0681
.1073
-0.0430
.0065
.0531
.0371
.0871
.1349
.1781
.2143
.2417
.2587
.2645
-0.0742
- .0238
.0390
.0823
.1184
.1452
.1613
.1658
.1582
.1392
.1098
.0716
.0268
.0256
.0710
.1097
.1394
.1585
.1657
.1607
.1437
.1157
.0785
.0341
.0943
.1279
.1519
.1651
.1667
.1566
.1355
.1044
.0650
.0198
.1383
.1595
.1698
.168h
.1558
.1324
.0994
.0587
.0124
-
-0.0589
- .0089
-
.0293
.0792
.1271
.1706
- .20.72
- .2348
- .2521
- .2579
-
.0219
.0719
.1203
.1644
.2015
.2297
.2472
.2532
-
.0147
.0652
.1143
.1592
.1972
.2260
.2440
.2502
-0.0892
- .0383
.0125
.0601
-0.1042
- .0527
- .0004
.0495
.1015
.1342
.1563
.1664
.1639
.1489
.1223
.0858
.0416
.0938
.1297
.1549
.1680
.1680
.1549
.1297
.0938
.0495
-
.0076
.0588
.1090
.1550
.1940
.2238
.2424
.2487
-
.0004
.0527
.1042
.1517
.1921
.2229
-.2422
- .2488
-0.1194
- .0673
- .0134
.0390
.0863
.1256
.1542
.1703
.1730
.1619
.1379
.1024
.0578
.0070
-
.0467
.0999
.1492
.1912
.2234
.2435
.2504
-0.1350
- .0823
- .0266
.0284
-0.1514
- .0980
- .0404
-0.1688
- .1147
- .0549
.0175
.0718
.1186
.1548
.1778
.1862
.1793
.1576
.1226
.0766
.0229
.0062
.0644
.1156
.1561
.1832
.1947
.1901
.1695
.1345
.0875
.0318
.0790
.1219
.1541
.1736
.1790
.1699
.1471
.1119
.0668
.0147
-
.0408
.0961
.1475
.1915
.2252
.2464
.2536
-
.0348
.0926
.1466
.1930
.2286
.2510
.2586
-
.0286
.0895
.1465
.1957
.2335
.2574
.2655
-0.1877
- .1328
- .0707
- .0059
.0568
.1129
.1583
.1898
.2049
.2028
.1834
.1482
.0998
.0416
-
.0220
.0866
.1473
.1999
.2404
.2659
.2746
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
0
z
I0.
to
0
CO
a
CO
CO
00
-1
CO
00
on
1
0
000-'o-'totoCOCOaa0000,-.]-I0000000
O
001001004100010000010010100010
111111
88888888SS88S8S888888
11111
I
I
I
I
I
0-'
0)
0-'
I
I
II II
tototot.'
51 -101000-0--'00100000-Il-'01000-10000
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0-000000100 COO a a 0100 00000-10000'
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II
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COCOa a 001 to CO C)
0000000000000
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to
01
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0500sto
to to to to 000to ;- to to
to
I
O
.I
II
CO
I
III
-.1
liii
I
CO
I
00000000000000000000000
0II
0
0000010 a 000)00001-ISo to a 0 a
0000 - -'0000 -0 - CO-I a
to Co a 0.40000
sooio a coo 0000)0050 Go a 0 a 4 00
oto
t-'oo to 0000055to to to to to
t-'
to to CO 0-'-4 CO 0008010-4 COCOS 010500000
to -40000010-40
1-' 01 00 .00000 tot-' SCO 1' S
4 COO a 0)0
CO
CO
to
CO
a
0
CO
0
0100)5010000)00005
0000000 toCOOto to 5000550
to to to
0
CO
tO
00
CO
CO
C'
CO
a
a
a
to
CO
00
0
CO
/
/
APPENDIX C
Table of Functions Pd and P'd
0.00
forra
1.00
0.08
0.06
0.14
0.12
0.10
0.18
0.16
0.20
0.24
0.22
Zfor F'a
0.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
0.04
0.02
-
-
0.0000
.0154
.0285
.0393
.0468
.0547
.0595
.0157
.0289
.0400
.0487
.0557
.0606
-.0626
-.0637
-
-
.0640
.0639
.0625
.0598
.0560
.0512
.0455
.0391
.0320
.0244
.0165
.0083
.0000
0.0000
.0159
0.0000
- .0161
.0294
.0406
.0497
.0567
.0617
-. .0299
.0413
.0506
.0577
.0630
-.0420
-.0650
--.0663
-.0677
0.0000
-
.0652
.0652
.0638
.0611
.0573
.0523
.0465
.0399
.0327
.0250
.0170
.0085
.0000
0.0000
-
--
.0666
.0666
.0652
.0624
.0585
.0535
.0476
.0409
.0335
.0256
.0173
.0087
.0000
-
.0680
.0680
.0666
.0638
.0598
.0548
.0487
.0419
.0343
.0262
.0177
.0089
.0000
- .0184
- .0303
- .0515
- .0588
- .0642
-
.0694
.0695
.0681
.0653
.0612
.0561
.0499
- - .0429
-
.0352
.0269
.0181
.0092
.0000
0.0000
- .0166
- .0309
- .0428
- .0524
- .0600
- .0655
- .0691
- .0709
- .0711
- .0697
- .0668
- .0627
- .0574
- .0511
- .0440
- .0360
- .0276
- .0186
- .0094
.0000
0.0000
- .0189
-.0314
-
.0436
.0534
.0612
.0669
.0706
.0725
.0727
.0713
.0684
.0642
.0588
.0524
.0451
.0370
.0283
.0191
.0096
.0000
0.0000
- .0172
--.0320
- .0444
- .0545
- .0624
- .0683
0.0000
- .0175
- .0325
- .0452
- .0556
- .0637
- .0698
0.0000
- .0178
- .0332
- .0461
- .0567
- .0651
- .0713
-.0721
-.0738
-.0755
-
-
-
.0741
.0744
.0730
.0701
.0658
.0603
.0537
.0462
.0379
.0290
.0196
.0099
.0000
.0758
.0761
.0747
.0718
.0675
.0619
.0551
.0474
.0389
.0298
.0201
.0101
.0000
.0777
.0780
.0766
.0736
.0692
-.0635
-
.0566
.0487
.0400
.0306
.0207
.0104
.0000
0.0000
- .0181
- .0338
-.0470
-
.0579
.0665
.0730
.0772
.0795
.0799
.0786
.0756
.0711
.0652
.0582
.0501
.0411
.0315
.0213
.0107
.0000
0.0000
- .0185
- .0345
- .0480
- .0592
- .0681
- 0747
-.0791
-
.0815
.0820
.0806
.0776
.0730
.0670
.0598
.0515
.0423
.0324
.0219
.0110
.0000
-
0.0000
1.00
.0188
.0352
.0491
.0605
.0696
.0765
.0811
.0836
.0842
.0828
.0797
.0750
.0689
.0615
.0530
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
- 0435
- .0333
- .0225
- .0114
.0000
APPENDIX C (Continued)
Table of Functions ra and r'd
0.26
0.28
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
forr"..
1.00
0.48
0.50
Vforr'a
.
0.0000
0
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
0.46
-
.0192
.0359
.0501
.0619
.0713
.0784
.0832
.0858
.0864
.0851
.0820
.0772
.0709
.0633
.0546
.0449
.0343
.0232
-.0117
.0000
0.0000
- .0196
- .0367
.0513
-
.0634
.0731
.0804
.0854
.0882
.0888
0.0000
- .0200
- .0375
- .0525
- .0650
-
.0749
.0825
.0877
.0906
.0914
0.0000
-
.0205
.0384
.0538
.0666
.0769
.0847
.0901
-.0932
- .0941
-.0875
-.0901
-.0928
-
-
-
.0843
.0795
.0730
.0652
.0562
.0462
.0354
.0239
.0121
.0000
.0869
.0819
.0753
.0673
.0580
.0477
.0365
.2047
.0125
.0000
.0895
.0844
.0777
.0694
.0599
.0492
.0377
.0255
.0129
.0000
0.0000
- .0209
- .0393
- .0551
- .0683
- .0790
- .0871
- .0927
- .0960
- .0969
- .0956
- .0923
- .0871
- .0802
- .0717
- .0619
- .0509
- .0390
- .0264
- .0133
.0000
0.0000
- .0214
- .0403
- .0566
- .0702
- .0812
- .0896
- .0955
- .0989
- .0999
0.0000
- .0220
- .0413
- .0581
0.0000
- .0225
- .1020
- .1031
-
.0424
.0597
.0742
.0861
.0952
.1016
.1053
.1066
0.0000
- .0231
- .0436
- .0614
- .0765
- .0887
- .0982
- .1049
- .1089
- .1102
-
-.0987
-.1019
-.1054
-.1091
-.1131
-
-
-
-
-
.0953
.0900
.0829
.0741
.0640
.0527
.0403
.0273
.0138
.0000
-.0722
- .0836
- .0923
-.0984
.0985
.0931
.0857
.0767
.0662
.0545
.0418
.0283
.0143
.0000
.1019
.0963
.0888
.0795
.0686
.0565
.0433
.0293
.0148
.0000 I
.1056
.0998
.0920
.0824
.0712
.0586
.0450
.0305
.0154
.0000
0.0000
.0238
.0449
.0633
.0789
.0916
.1015
.1085
.1127
.1142
.1095
.1036
.0955
.0856
.0740
.0609
.0467
.0317
.0160
.0000
0.0000
- .0244
- .0462
- .0653
- .0814
- .0947
- .1050
- .1123
- .1168
- .1184
- .1173
- .1137
- .1076
- .0993
- .0890
- .0770
- .0634
- .0486
- .0329
- .0166
.0000
0.0000
-. .0252
-
.0477
.0674
.0842
.0980
.1087
.1164
.1212
.1230
-
0.0000
.0260
.0493
.0697
.0872
.1015
.1128
.1209
.1259
.1279
-.1219
-.1269
-
-
.1182
.1119
.1034
.0927
.0802
.0661
.0507
.0343
.0173
.0000
.1231
.1166
.1077
.0966
.0836
- .0689
- .0529
- .0358
- .0181
.0000
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
0
APPENDIX C (Continued)
Table of Functions ra and f'a
0.52
0.50
0.54
0.56
0.58
0.0000
0.0000
0.62
0.60
0.66
0.64
0.68
0.70
0.72
0.74
0.0000
- .0421
- .0810
- .1163
- .1474
0.0000
- .0449
Zfor P'a
for Pa N
0.00
.05
.10
.15
.20
.25
.20
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
0.0000
- .0260
- .0493
- .0697
- .0872
- .1015
- .1128
- .1209
- .1259
- .1279
- .1269
- .1231
- .1166
- .1077
- .0966
- .0836
-.0689
- .0529
- .0358
- .0181
.0000
0.0000
-
.0268
.0510
.0722
.0904
.1054
.1172
.1257
.1311
.1332
.1322
.1284
.1217
.1125
.1009
.0874
.0721
.0553
.0375
.0189
.0000
0.0000
- .0278
- .0528
- .0749
- .0039
- .1096
- .1220
- .1310
- .1366
- .1390
- .1381
- .1341
- .1272
- .1176
- .1056
- .0915
-
-.0754
-.0791
- .0579
- .0393
- .0198
.0000
- .0608
- .0412
- .0208
.0000
.0288
.0548
.0778
.0977
.1142
.1272
.1367
.1427
.1453
.1444
.1404
.1332
.1233
.1107
.0059
0.0000
0.0000
- .0340
- .0650
- .0928
- .1171
0.0000
0.0000
- .0375
0.0000
- .1163
- .1008
0.0000
- .0311
- .0594
- .0846
- .1064
- .1246
- .1391
- .1499
- .1567
- .1598
- .1591
- .1548
- .1471
- .1363
- .1225
- .1062
-
-.0832
-.0877
-.0926
-.0981
-.1042
-.1112
-.1190
-.1280
-.1383
- .0639
- .0433
- .0219
- .0674
- .0457
- .0231
- .0754
- .0511
- .0258
- .0801
- .0544
- .0275
.0000
.0000
.0000
- .0855
- .0580
- .0293
.0000
- .0915
- .0621
- .0314
.0000
- .0712
- .0483
- .0244
.0000
- .0985
- .0668
- .0338
.0000
- .1065
- .0723
- .0365
.0000
-
.0299
.0570
.0811
.1018
.1191
.1329
.1430
.1494
.1522
.1514
.1473
.1398
-. .1295
.0325
.0621
.0885
.1115
.1307
.1461
.1575
.1648
.1682
.1676
.1632
.1552
.1438
.1293
.1121
-
.1374
.1537
.1659
.1738
.1775
.1770
.1725
.1641
.1521
.1369
.1187
-
.0356
.0683
.0977
.1233
.1449
.1623
.1753
.1839
.1879
.1875
.1828
.1741
.1615
.1454
.1261
-
.0721
.1031
.1303
.1534
.1720
-.. .1859
-
.1952
.1996
.1994
.1945
.1853
.1720
.1549
.1345
-
.0396
.0762
.1093
.1383
.1629
.1829
.1980
.2080
.2129
.2128
.2078
.1081
.1839
.1657
.1439
.0000
-. .1739
-
.1954
.2117
.2226
.2281
.2282
.2229
.2126
.1975
.1781
.1547
-
.0865
.1244
.1579
.1865
.2098
.2275
.2395
.2456
.2459
.2404
.2294
.2133
.1924
.1672
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
I-I.---
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/
APPENDIX C (Continued)
Table of Functions F4 and F'4
1.04
1.08
1.16
1.12
1.20
1.24
1.28
1.32
1.36
1.40
1.44
1.48
1.52
0.0000
.0211
.0439
.0674
.0908
.1132
.1336
.1514
.1656
.1759
.1816
.1825
.1784
.1693
.1554
.1371
.1148
.0892
.0610
.0309
.0000
0.0000
.0182
.0382
.0591
.0801
.1003
.1190
.1352
.1485
.1581
.1637
.1648
.1614
.1535
.1411
.1246
.1045
.0812
.0555
.0282
.0000
0.0000
.0159
.0336
.0523
.0713
.0898
.1069
.1220
.1345
.1436
.1490
.1504
.1476
.1406
.1294
.1144
.0960
.0747
.0511
.0260
.0000
0.0000
.0139
.0296
.0466
.0639
.0809
.0968
.1110
.1227
.1315
.1368
.1384
.1361
.1299
.1197
.1060
.0890
.0693
.0474
.0241
.0000
0.0000
.0122
.0263
.0417
.0576
.0734
.0883
.1016
.1128
.1212
.1265
.1283
.1264
.1208
.1116
.0989
.0831
.0648
.0444
.0225
.0000
for F'4
Fi
00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
0.0000
.2459
.4878
.7195
.9348
1.1283
1.2947
1.4297
1.5297
1.5921
1.6150
1.5979
1.5409
1.4455
1.3140
1.1495
.9563
.7391
.5033
.2548
.0000
0.0000
.1196
.2385
.3532
.4606
.5579
.6421
.7110
.7026
.7955
.8086
.8014
.7740
.7271
.6617
.5794
.4824
.3730
.2541
.1287
.0000
0
.0775
.1553
.2310
.3025
.3677
.4245
.4714
.5070
.5300
.5398
.5360
.5185
.4877
.4444
.3895
.3246
.2511
.1712
.0867
.0000
0.0000
.0565
.1137
.1699
.2234
.2725
.3157
.3516
.3791
.3973
.4055
.4034
.5909
.3682
.3358
.2947
.2457
.1903
.1297
.0657
.0000
0.0000
.0438
.0886
.1331
.1758
.2153
.2503
.2796
.3023
.3176
.3249
.3238
.3142
.2964
.2707
.2378
.1985
.1538
.1049
.0532
.0000
0.0000
.0353
.0719
.1086
.1441
.1772
.2067
.2317
.2512
.2645
.2712
.2708
.2633
.2487
.2275
.2000
.1671
.1295
.0884
.0448
.0000
0.0000
.0292
.0599
.0910
.1213
.1498
.1754
.1973
.2145
.2265
.2327
.2329
.2268
.2146
.1965
.1730
.1446
.1122
.0766
.0388
.0000
.0247
.0509
.0778
.1042
.1292
.1519
.1715
.1870
.1980
.2040
.2045
.1996
.1891
.1734
.1528
.1278
.0992
.0678
.0344
.0000
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
APPENDIX C (Continued)
Table of Functions Fi and r'a
1.52
1.56
1.60
1.64
1.68
1.72
1.76
1.80
1.84
1.88
1.92
1.96
2.00
0.00
0.0000
.05
.10
.15
.01.22
0.0000
.0107
.0234
.0375
.0522
.0669
.0809
.0936
.1042
.1124
.1177
.1196
.1181
.1131
.1046
.0928
.0781
.0609
.0417
.0212
.0000
0.0000
.0095
.0209
.0338
.0475
.0612
.0745
.0866
.0968
.1048
.1100
.1121
.1110
.1064
.0986
0.0000
.0083
.0187
.0305
.0433
.0563
.0688
.0804
0.0000
.0073
.0167
.0276
.0395
.0518
.0638
.0749
.0846
.0922
.0974
.0998
.0992
0.0000
.0064
.0149
.0250
.0362
.0478
.0593
.0700
.0794
.0869
.0921
.0947
.0944
.0910
.0847
.0756
.0639
.0500
.0343
.0175
.0000
0.0000
.0056
.0133
.0227
.0332
.0442
.0553
.0656
.0748
.0822
.0874
.0901
.0900
.0870
.0811
.0725
.0613
.0480
.0330
.0168
.0000
0.0000
.0048
.0118
.0205
.0304
.0410
.0516
.0616
.0706
.0779
.0832
.0860
.0861
.0835
.0780
.0698
.0591
.0463
.0318
.0162
.0000
0.0000
.0041
.0104
.0185
.0279
.0380
.0482
.0580
.0668
.0741
.0794
.0823
.0827
.0803
.0751
.0673
.0571
.0448
.0308
.0157
.0000
0.0000
.0035
.0092
.0167
.0255
.0352
.0451
.0547
.0633
.0706
.0759
.0790
.0795
.0774
.0725
.0651
.0553
.0434
.0299
.0152
.0000
0.0000
.0029
.0080
.0150
.0234
.0326
.0423
.0516
.0601
.0673
.0727
.0759
0.0000
.0023
0.0000
.0018
.0059
.0119
.0194
.0280
.0371
.0461
.0545
.0616
.0672
.0706
.0717
.0703
.0663
.0598
.0509
.0401
.0277
.0141
.0000
for r1
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
-'for F'a
.0263
.0417
.0576
.0734
.0883
.1.016
.1128
.1212
.1265
.1283
.1264
.1208
.1116
.0989
.0831
.0648
.0444
.0225
.0000
.0876
.0738
.0576
.0395
.0201
.0000
.0903
.0981
.1033
.1056
.1047
.1006
.0933
.0831
.0700
.0547
.0375
.0191
.0000
.O55
.0888
.0791
.0668
.0522
.0358
.0182
.0000
.07.67
.0748
.0702
.0631
.0537
.0422
.0290
.0148
.0000
.0069
.0134
.0213
.0303
.0396
.0488
.0572
.0644
.0698
.0731
.0741
.0724
.0682
.0614
.0522
.0411
.0283
.0144
.0000
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
API'ENDIX C (Continued)
Table of Functions F0 and F'0
X.INfl
2.00
2.08
0.0000
.0018
.0059
.0119
.0194
.0280
.0371
.0461
.0545
.0616
.0672
.0706
.0717
.0703
.0663
.0598
.0509
.0401
.0277
.0141
.0000
0.0000
.0008
.0040
.0091
.0160
.0240
.0326
.0413
.0496
.0567
.0624
.0661
.0676
.0666
.0630
.0570
.0487
.0384
.0266
.0136
.0000
2.16
2.24
2.32
2.40
2.48
2.56
2.64
2.72
0.0000
- .0047
- .0067
- .0060
- .0029
.0025
.0095
.0177
.0263
.0347
.0423
.0483
.0524
.0540
.0530
.0493
.0431
.0345
.0241
.0124
.0000
0.0000
- .0055
- .0082
2.80
2.88
2.96
0.0000
- .0071
- .0114
- .0126
- .0108
- .0063
.0006
0.0000
- .0081
- .0132
- .0151
- .0138
.0091
.0186
.0283
.0374
.0450
.0506
.0536
.0537
.0508
.0449
.0363
0.255
.0131
.0000
.0061
.0160
.0263
.0360
.0444
.0506
.0542
.0547
.0520
.0461
.0374
.0263
.0136
.0000
Zfor F'
for
0.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
0.0000
- .0001
.0022
.0066
.0128
.0203
.0286
.0371
.0452
.0525
.0583
.0623
.0641
.0635
.0604
.0548
.0470
.0371
.0257
.0131
.0000
0.0000
- .0009
.0006
.0043
.0099
.0169
.0249
.0333
.0414
.0487
.0548
.0590
.0612
.0609
.0582
.0531
.0456
.0361
.0250
.0128
.0000
0.0000
- .0017
- .0009
.0021
.0072
.0138
.0215
.0298
.0379
.0454
.0517
.0582
.0587
.0588
.0565
.0517
.0445
.0354
.0245
.0126
.0000
0.0000
- .0025
- .0024
.0000
.0046
.0109
.0184
.0265
.0347
.0424
.0489
.0538
.0566
.0571
.0551
.0506
.0438
.0348
.0242
.0124
.0000
0.0000
- .0032
- .0038
- .0020
.0021
.0080
.0153
.0234
.0318
.0396
.0464
.0517
.0549
.0558
.0541
.0499
.0433
.0345
.0240
.0123
.0000
0.0000
- .0039
- .0053
- .0040
- .0004
.0052
.0124
.0205
.0290
.0371
.0443
.0499
.0535
.0547
.0534
.0495
.0431
0344
.0240
.0123
.0000
- .0081
- .0054
- .0004
.0066
.0148
.0237
.0325
.0405
.0470
.0515
.0536
.0529
.0495
.0434
.0349
.0244
.0126
.0000
0.0000
-
.0063
.0097
.0103
.0080
.0032
.0036
.0120
.0212
.0304
.0389
.0459
.0509
.0534
.0532
.0500
.0440
.0354
.0248
.0128
.0000
- .0095
- .0027
1.00
.95
.90
.85
.80
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
0
;
00
e
00
00
00
00
0
00
00
00
0
00
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61
OREGON STATE COLLEGE
ENGINEERING EXPERIMENT STATION
CORVALLIS, OREGON
LIST OF PUBLICATIONS
BulletinsNo.
1.
No. 2.
Preliminary Report on the Control of Stream Pollution in Oregon, by C. V.
Langton and H. S. Rogers. 1929.
Fifteen cents.
A Sanitary Survey of the Willamette Valley, by H. S. Rogers, C. A. Mockmore, and C. D. Adams.
Forty cents.
1930.
3.
The Properties of Cement.Sawdust Mortars, Plain, and with Various Admix.
tures, by S. H. Graf and R. H. Johnson. 1930.
No. 4.
Interpretation of Exhaust Gas Analyses, by S. H. Graf, G. W. Gleeson, and
W. H. Paul. 1934.
No. 5.
Boiler-Water Troubles and Treatments with Special Reference to Problems in
Western Oregon, by R. E. Summers. 1935.
No.
Twenty cents.
Twenty-five cents.
None available.
No.
6. A Sanitary Survey of the Willamette River from Seliwood Bridge to the
Columbia, by G. W. Gleeson. 1936.
Twenty-five cents.
Industrial and Domestic Wastes of the Willamette Valley, by G. W. Gleeson
and F. Merryfield. 1936.
Fifty cents.
No. 8. An Investigation of Some Oregon Sands with a Statistical Study of the Pre.
dictive Values of Tests, by C. E. Thomas and S. H. Graf. 1937.
Fifty cents.
No. 9. Preservative Treatments of Fence Posts.
1938 Progress Report on the Post Farm, by T. J. Starker, 1938.
Twenty-five cents.
Yearly progress report, 9-A, 9-B, 9-C, 9-D, 9-E, 9-F.
Fifteen cents each.
No. 10. Precipitation-Static Radio Interference Phenomena Originating on Aircraft, by
E. C. Starr, 1939.
No. 7.
Seventy-five cents.
No. 11.
Electric Fence Controllers with Special Reference to Equipment Developed
for Measuring Their Characteristics, by F. A. Everest. 1939.
No. 12.
Mathematics of Alignment Chart Construction without the Use of Deter-
No. 13.
Oil Tar Creosote for Wood Preservation, by Glenn Voorhies, 1940.
Forty cents.
minants, by J. R. Griffith.
Twenty-five cents.
1940.
No. 14.
Twenty-five cents.
Optimum Power and Economy Air-Fuel Ratios for Liquefied Petroleum Gases,
by W. H. Paul and M. N. Popovich. 1941.
Twenty.five cents.
No. 15.
Rating and Care of Domestic Sawdust Burners, by E. C. Willey.
No. 16.
The Improvement of Reversible Dry Kiln Fans, by A. D. Hughes. 1941.
No. 17.
An Inventory of Sawmill Waste in Oregon, by Glenn Voorhies. 1942.
No. 18.
The Use of Fourier Series in the Solution of Beam Problems, by B. F. Ruff-
No. 19.
No. 20.
No. 21.
1941.
Twenty-five cents.
Twenty-five cents.
Twenty-five cents.
ncr.
1944.
Fifty cents.
1945 Progress Report on Pollution of Oregon Streams, by Fred Merryfield and
W. G. Wilmot. 1945.
Forty cents.
The Fishes of the Willamette River System in Relation to Pollution, by R. E.
Dimick and Fred Merryfield. 1945.
Forty cents.
The Use of the Fourier Series on the Solution of Beam-Column Problems,
by B. F. Ruffoer.
Twenty-five cents.
1945.
62
FOURIER SERIES IN BEAM-COLUMN PROBLEMS
63
CircularsNo. 1.
No.
2.
No. 3.
A Discussion of the Properties and Economics of Fuels Used in Oregon, by
C. E. Thomas and G. D. Keerins. 1929.
Twenty-five cents.
Adjustment of Automotive Carburetors for Economy, by S. H. Graf and G. W.
Gleeson. 1930.
None available.
Elements of Refrigeration for Small Commercial Plants, by W. H. Martin.
1935.
None available.
Some Engineering Aspects of Locker and Home Cold.Storage Plants, by W. H.
Martin. 1938.
Twenty cents.
No. 4.
No.
5.
Refrigeration Applications to Certain Oregon Industries, by W. H. Martin.
1940.
No. 6.
Twenty-five cents.
The Use of a Technical Library, by W. E. Jorgensen.
1942.
Twenty-five cents.
Saving Fuel in Oregon Homes, by E. C. Willey. 1942.
Twenty.five cents.
No 8. Technical Approach to the Utilization of Wartime Motor Fuels, by W. H. Paul.
1944.
Twenty.five cents.
No.
7.
ReprintsNo.
1.
Methods of Live Line Insulator Testing and Results of Tests with Different
Instruments, by F. 0. McMillan. Reprinted from 1927 Proc. N. W. Elec.
Lt. and Power Assoc.
Twenty cer.ts.
No. 2.
Some Anomalies of Siliceous Matter in Boiler \Vater Chemistry, by R. E.
Summers. Reprinted from Jan. 1935, Combustion.
Ten cents.
No. 3. Asphalt Emulsion Treatment Prevents Radio Interference, by F. 0. McMillan.
Reprinted from Jan. 1935, Electrical West.
No. 4.
None available.
Some Characteristics of A.0 Conductor Corona, by F. 0. McMillan.
from Mar. 1935, Electrical Engineering.
None available.
Reprinted
No. 5. A Radio Interference Measuring Instrument, by F. 0. McMillan and H. G.
Barnett. Reprinted from Aug. 1935, Electrical Engineering.
No. 6.
No.
7.
No.
8.
No. 9.
Ten cents.
Water-Gas Reaction Apparently Controls Engine Exhaust Gas Composition, by
G. %V. Gleeson and W. H. Paul. Reprinted from Feb. 1936, National
Petroleum News.
Ten Cents.
Steam Generation by Burning Wood, by R. E. Summers. Reprinted from
April 1936, Heating and Ventilating.
Ten cents.
The Piezo Electric Engine Indicator, by W. H. Paul and K. R. Eldredge.
Reprinted from Nov. 1935, Oregon State Technical Record.
Ten cents.
Humidity and Low Temperatures, by W. H. Martin and E. C. Willey. Re.
printed from Feb. 1937, Power Plant Engineering.
None available.
No. 10.
Heat Transfer Efficiency of Range Units, by W. J. Walsh.
Aug. 1937, Electrical Engineering.
None available.
Reprinted from
Design of Concrete Mixtures, by I. F. Waterman. Reprinted from Nov. 1937,
Concrete.
None available.
No. 12. Water-wise Refrigeration, by W. H. Martin and R. E. Summers. Reprinted
from July 1938, Power.
None available.
No. 13. Polarity Limits of the Sphere Gap, by F. 0. McMillan. Reprinted from Vol.
58, A.I.E.E. Transactions, Mar. 1939.
Ten cents.
No. 14. Influence of Utensils on Heat Transfer, by W. G. Short. Reprinted from
Nov. 1938, Electrical Engineering.
Ten cents.
No. 15. Corrosion and Self-Protection of Metals, by R. E. Summers. Reprinted from
Sept. and Oct. 1938, Industrial Power.
Ten cents.
No. 11.
ENGINEERING EXPERIMENT STATION BULLETIN 21
64
No. 16.
Monocoque Fuselage Circular Ring Analysis, by B. F. Ruffner.
No. 18.
Fuel Value of Old.Growth vs. Second.Growth Douglas Fir, by Lee Gable.
No. 19.
Stoichiometric Calculations of Exhaust Gas, by G. W. Gleeson and F. W.
Woodfield, Jr. Reprinted from November 1, 1939, National Petroleum
No. 20.
The Application of Feedback to Wide.Band Output Amplifiers, by F. A.
Reprinted
from Jan. 1939, Journal of the Aeronautical Sciences.
Ten cents.
No. 17. The Photoelastic Method as an Aid in Stress Analysis arid Structural Design,
by B. F. Ruffner. Reprinted from Apr. 1939, Aero Digest.
Ten cents.
Reprinted from June 1939, The Timberrnan.
Ten cents.
News.
No. 21.
No. 22.
No. 23.
No. 24.
No. 25.
No. 26.
Ten cents.
Everest and H. R. Johnston. Reprinted from February 1940, Proc. of the
Institute of Radio Engineers.
Ten cents.
Stresses Due to Secondary Bending, by B. F. Ruffner. Reprinted from Proc.
of First Northwest Photoelasticity Conference, University of Washington,
March 30, 1940.
Ten cents.
Wall Heat Loss Back of Radiators, by E. C. Willey. Reprinted from No.
vember 1940, Heating and Ventilating.
Ten cents.
Stress Concentration Factors in Main Members Due to Welded Stiffeners, by
W. R. Cherry. Reprinted from December, r941, The Welding Journal,
Research Supplement.
Ten cents.
Horizontal.Polar.Pattern Tracer for Directional Broadcast Antennas, by F. A.
Everest and XV. S. Pritchett. Reprinted from May, 1942, Proc. of The
Institute of Radio Engineers.
Ten cents.
Modern Methods of Mine Sampling, by R. K. Meade. Reprinted from January, 1942, The Compass of Sigma Gamma Epsilon.
Ten cents.
Broadcast Antennas and Arrays. Calculation of Radiation Patterns; Impedance Relationships, by Wilson Pritchett.
September, 1944, Communications.
Fifteen cents.
Reprinted from August and
THE ENGINEERING EXPERIMENT STATION
Administrative Officers
W. L. MARKS, President, Oregon State Board of Higher Education.
F. M. HUNTER, Chancellor, Oregon State System of Higher Education.
A. L. STRAND, President, Oregon State College.
G. W. GLEESON, Dean, School of Engineering.
D. M. GOODE, Editor of Publications.
S. H. GRAF, Director, Engineering Experiment Station.
Station Staff
A. L. ALBERT, Communication Engineering.
P. M. DUNN, Forestry.
G. S. FEIKERT, Radio Engineering.
G. W. GLEESON, Chemical Engineering.
BURDETTE GLENN, Highway Engineering.
G. W. HOLCOMB, Structural Engineering.
C. V. LANCTON, Public Health.
F. 0. MCMILLAN, Electrical Engineering.
W. H. MARTIN, Mechanical Engineering.
FRED M ERRYFIELD, Sanitary Engineering.
C. A. MOCXMORE, Civil and Hydraulic Engineering.
W. H. PAUL, Automotive Engineering.
B. F. RUFFNER, Aeronautical Engineering.
*A. W. SCFILECHTEN, Mining and Metallurgical Engineering.
M. C. SHEELY, Shop Processes.
Electric Space Heating.
tE. C. STARR, Electrical Engineering.
C. E. THOMAS, Engineering Materials.
GLENN VoosH IFS, Wood Products.
E. C. WILLEY, Air Conditioning.
Technical Counselors
R. H. BALDOCK, State Highway Engineer, Salem.
IVAN BLOCH, Chief, Division of industrial and Resources Development, Bonne-
ville Power Administration, Portland.
R. R. CLARK, Designing Engineer, Corps of Engineers, Portland District,
Portland.
DAVID DON, Chief Engineer, Public Utilities Commissioner, Salem.
C. B. MCCULLOUGH, Assistant State Highway Engineer, Salem.
PAUL B. MCKEE, President, Portland Gas and Coke Company, Portland.
B. S. MORRO\V, Engineer and General Manager, Department of Public Utilities
and Bureau of Water Works, Portland.
F. W. LTBBEY, Director, State Department of Geology and Mineral Industries,
Portland.
J. H. POLHEMUS, President, Portland General Electric Company, Portland.
S. C. SCHVARZ, Chemical Engineer, Portland Gas and Coke Company, Portland.
J. C. STEVENS, Consu1[ting Civil and Hydraulic Engineer, Portland.
C. E. STRICKLIN, State Engineer, Salem.
S. N. WYCKOFF, Director, Pacific Northwest Forest and Range Experiment
Station, U. S. Department of Agriculture, Forest Service, Portland.
On leave of absence for military or civilian war service.
Oregon State College
Corvallis
RESIDENT INSTRUCTION
Liberal Arts and Sciences
THE Lowrx DIVISION (Junior Certificate)
SCHOOL op SCIENCE (B.A., B.S., MA., M.S., Ph.D. degrees)
The Professional Schools
ScHooL or AGRICULTURE (B.S., B.Agr., M.S., Ph.D. degrees)
DivisioN OF BUSINESS AND INDUSTRY (B.A., B.S., B.S.S. degrees)
SCHOOL OF EDUCATION (B.A., B.S., Ed.B., M.A., M.S., Ed.M.,
Ed.D. degrees)
SCHOOL OF
ENGINEERING AND INDUSTRIAL ARTS (B.A., B.S.,
B.I.A., M.A., M.S., Ch.E, C.E., E.E., M.E., Met.E., Min.E.
Ph.D. degrees)
SCHOOL OF
SCHOOL OF
FORESTRY (B.S., B.F., M.S., M.F., F.E. degrees)
HOME EcoNoMIcs (B.A., B.S., M.A., M.S., Ph.D.
degrees)
SCHOOL OF PHARMACY (B.A., B.S., M.A., M.S. degrees)
The Graduate Division (M.A., M.S., Ed.M., M.F., Ch.E., C.E., E.E.,
F.E., M.E., Met.E., Min.E., Ed.D., Ph.D. degrees)
The Summer Sessions
The Short Courses
RESEARCH AND EXPERIMENTATION
The General Research Council
The Agricultural Experiment Station
The Central Station, Corvallis
The Union, Moro, Hermiston, Talent, Astoria, Hood River,
Pendleton, Medford, and Squaw Butte Branch Stations
The Northrup Creek, Klamath, Malheur, and Red Soils Experimental Areas
The Engineering Experiment Station
The Oregon Forest Products Laboratory
EXTENSION
Federal Cooperative Extension (Agriculture and Home Economics)
General Extension Division