I LL IN I
S
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
PRODUCTION NOTE
University of Illinois at
Urbana-Champaign Library
Large-scale Digitization Project, 2007.
~>
UNIVERSITY OF ILLINOIS BULLETIN
SISSUD WMmKT
V1. XXVIII October 14, 1930 No.7 -
[Entered as senad-clase matter December 11, 1912, at the post office at Urbana, Illinois, under
the Act of August 2, 1912. Acceptance for mailing at the pecal rate of postage provided
for in section 1103, Act of Oetober 8, 1917, authorized July 31, 1918.]
THE COLUMN ANALOGY
BY
HARDY CROSS
BULLETIN No. 215
ENGINEERING EXPERIMENT STATION
PUWaB.is B yr .ex stvrmieM or Ii4tol, UeAxA
Piacs: Fodft CNa .
HT- HE Engineering Experiment Station was established by act
of the Board of Trustees of the University of Illinois on De-
cember 8, 1903. It is the purpose of the Station to conduct
investigations and, make studies of importance to the engineering,
manufacturing, railway, mining, and other industrial interests of the
State.
The management of the Engineering Experiment Station is vested
in an Executive Staff composed of the Director and his Assistant, the
Heads of the several Departments in the College of Engineering, and
the Professor of Industrial Chemistry. This Staff is responsible for
the establishment of general policies governing the work of the Station,
-including the -approval of material for publication. All members of
the teaching staff of the College are encouraged to engage in scientific
research, either directly or in cooperation with the Research Corps
composed of full-time research assistants, research graduate assistants,
and special investigators.
To render the results of its scientific investigations available to
the public, the Engineering Experiment Station publishes and dis-
tributes a series of bulletins. Occasionally it publishes circulars of
timely interest, presenting information of importance, compiled from
various sources which may not readily be accessible to the clientele
of the Station.
The volume and number at the top of the front cover page are
merely arbitrary numbers and refer to the general publications of the
University. Either above the title or below the seal is given the num-
ber of the Engineering Experiment Station bulletin or circular which
should be used in referring to these publications.
For copies of bulletins or circulars or for other information address
THE ENGINEERING EXPERIMENT STATION,
UNIVERSITY OF ILLINOIS,
URBANA, ILLINOIS
UNIVERSITY OF ILLINOIS
ENGINEERING EXPERIMENT STATION
BULLETIN No. 215
OCTOBER, 1930
THE COLUMN ANALOGY
ANALYSIS OF ELASTIC ARCHES AND FRAMES BY THE
GENERAL FORMULA FOR FLEXURE
BY
HARDY CROSS
PROFESSOR OF STRUCTURAL ENGINEERING
ENGINEERING EXPERIMENT STATION
PUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA'
UNIVER eY
"»'»"**", °;E's°;,
CONTENTS
I. INTRODUCTION . . . . . . . . . . .
1. Purpose of the Monograph . . . . . .
2. Validity of Analyses by the Theory of Elasticity
3. Acknowledgment . . . . . . . . .
PAGE
7
7
8
9
II. ANALYSES FOR FLEXURAL STRESS .
4. General Equation of Flexural Stress.
5. Form of Computation Recommended
6. Transformed Section . . . . .
III. THE COLUMN ANALOGY . . . . . .
7. Formulas Similar to Flexure Formula
8. Geometry of Continuity . . . .
9. The Analogy . . . . . . .
10. The Elastic Column and Its Load
11. Signs in the Column Analogy
Choice of Statically Determined Moments
Components and Direction of Indeterminate Forces
Application to Simple Cases . . . . . . .
Simple Numerical Examples . . . . . . .
Arch Analysis . . . .
Haunched Beams . . . . . . . . . .
Slopes and Deflections of Beams. . . . . .
Supports of the Analogous Column . . . . .
IV. EXTENSION OF THE ANALOGY
20. Introduction .
21. Internal Distortions
V. PHYSICAL CONSTANTS OF DEFORMATION FOR STRUCTURAL
MEMBERS . . . . . . . . . . . .
22. Nature of Physical Constants
23. Method of Determining Physical Constants .
24. Computation of the Constants . . . . . .
9
9
. . . . 14
18
. . . . 22
. . . . 22
. . . . 22
. . . . 25
. . . . 27
. . . . 28
. . . 57
. . . 57
. . . 57
4 CONTENTS (Concluded)
VI. APPLICATIONS OF THEOREM . . . . . . . . . 70
25. Fields of Application of Theorem . . . . 70
26. Methods of Analysis of Continuous Frames . . 71
27. General Equation of Displacements and Slope-
Deflection . . . . . . . . . . . . 72
VII. CONCLUSION . . . . . . . . . . . . . 75
28. Conclusion . . . . . . . . . . . . 75
LIST OF FIGURES
NO. PAGE
1. Example of an Unsymmetrical Section . . . . . . . . . . 15
2. Example of a Symmetrical Section, Unsymmetrically Loaded . . . . 17
3. Example of a Symmetrical Section, Symmetrically Loaded . . . . . 19
4. Segment of a Curved Beam . . . . . . . . . . . . . . 21
5. Ratio, r -= ..... ............. . 22
6. Closed Ring, Cut and Subjected to an Angle Change . . . . . . 22
7. Types of Analogous Column Sections . . . . . . . . . . . 26
8. Possible Moment Loads for Beam with Fixed Ends and for Rectangular Bent 30
9. Application of the Principle of Column Analogy to Simple Cases . . . 32
10. Rotation and Displacement of the Ends of a Member . . . . . . 34
11. Beam with Fixed Ends . . . . . . . . . . . . . . . 35
12. Rectangular Bent . . . . . . . . . . . . . . . . . 36
13. Bent with Monitor . . . . . . . . . . . . . . . . 39
14. Unsymmetrical Bent . . . . . . . . . . . . . . . . 40
15. Unsymmetrical Arch . . . . . . . . . . . . . . . . 42
16. Symmetrical Arch . . . . . . . . . . . . . . . . 46
17. Beam Symmetrically Haunched . . . . . . . . . . . . 49
18. Beam Unsymmetrically Haunched . . . . . . . . . . . . 50
19. Influence Lines for Shear and Moment in an Arch with Fixed Ends. . . 52
20. Types of Supports for Analogous Columns. . . . . . . . .. . 54
21. Types of Supports for Analogous Columns . . . . . . . .. . 56
22. Terms Used for Internal Distortions . . . . . . . . . . . 58
23. Experimental Determination of Physical Constants . . . . . . . 62
24. Flexural Distortion of a Beam . . . . . . . . . . . .. 63
25. Deformation Constants for Web Members in a Truss . . . . . . 65
26. Segment of a Curved Beam . . . . . . . . . . . . . . 66
27. Closed Elastic Ring . . . . . . . . . . . . . . . . 70
THE COLUMN ANALOGY
I. INTRODUCTION
1. Purpose of the Monograph.-The object of this bulletin is to
present some theorems dealing with the elastic analysis of continuous
frames. In ordinary cases these theorems are identical in form with
the theorems, with which every structural engineer is familiar, for
finding internal stresses in beams and struts.
The subject of structural mechanics is now experiencing demands
for greater precision by the very accurate analyses demanded in the
design of airplanes. In any case it is of the greatest importance to
isolate definitely those matters which are sources of uncertainty from
those which are certain and hence not proper fields of experiment.
Problems dealing with the analysis of restrained flexural mem-
bers-straight beams, bents, arches-occupy a large space in struc-
tural literature. The treatment often presented involves complicated
equations; in nearly all cases the method of solution is hard to
remember.
If the elastic properties of the different portions of the structure
are definitely known, the analysis of restrained members is essentially
a problem in geometry, because the member must bend in such a way
as to satisfy the conditions of restraint. The geometrical relations
involved are identical in algebraic form with the general formula for
determining fiber stress in a member which is bent.
Since the analysis of problems in flexure is a familiar procedure to
structural engineers, it is advantageous to state the relations involved
in the analysis of fixed-end beams, bents, and arches in terms of the
beam formula. The advantages for structural engineers are similar
to those which result from using the theorems of area-moments in
finding slopes and deflections of beams; in some respects the concepts
involved are identical, and the use of the beam formula in the analysis
of restrained members may be thought of as an extension of the prin-
ciples of area-moments. The general conception referred to in this
monograph as the "column analogy" includes the principles of area-
moments and also the conception of the conjugate beam.*
In this bulletin it is shown that bending moments in arches,
haunched beams, and framed bents may be computed by a procedure
*See H. M. Westergaard, "Deflection of Beams by the Conjugate Beam Method," Journal of
Western Society of Engineers, December, 1921.
ILLINOIS ENGINEERING EXPERIMENT STATION
analogous to the computation of fiber stresses in short columns subject
to bending, and that slopes and deflections in these structures may
be computed as shears and bending moments, respectively, on
longitudinal sections through such columns.
The theorem makes available for the analysis of plane elastic
structures the literature of beam analysis, dealing with the kern, the
circle of inertia, the ellipse of inertia, graphical computations of
moments and products of inertia, and conjugate axes of inertia.
Certain terms are .defined in such a way that the method is ex-
tended to include the effect of deformations due to longitudinal stress
and to shear in ribbed members, and to include trussed members.
The conceptions used in arch analysis by these methods make
possible a general statement of the relations of joint displacements to
joint forces, of which the familiar equation of slope-deflection* is a
special case, and hence make possible the convenient extension of
the method of slope-deflection, or of the theorem of three moments, to
include curved members and members of varying moment of inertia.
The method here presented has application in the fields both of
design and of research. In the field of design we use certain physical
properties of the materials, which are necessarily assumed. In re-
search we may either resort to the laboratory and study by empirical
methods the properties of the structure as a whole, or we may study
only the physical properties of the materials themselves, and depend
on the geometrical relations to determine the properties of the struc-
ture. It seems obvious that the geometrical relations are not them-
selves a proper subject for experimental research.
The relations pointed out in this bulletin have at first been care-
fully restricted to geometry, and the assumptions which are necessary
to apply this geometry to the design of structures are developed later
in the discussion.
2. Validity of Analyses by the Theory of Elasticity.-The mathe-
matical identity of the expressions for moment in an elastic ring and
for fiber stress in a column section has some value in considering in a
qualitative way the general validity of analyses based on the elastic
theory.
It appears at times that engineers are not altogether discriminat-
ing in considering the value of elastic analyses, and seem to hold that
one must either accept as precise the results of such analyses or reject
entirely their conclusions.
*"Analysis of Statically Indeterminate Structures by the Slope-Deflection Method," Univ. of
Ill. Eng. Exp. Sta. Bul. 108, 1918.
THE COLUMN ANALOGY
Now no one but a novice accepts without discrimination the re-
sults of the beam formula. It is open to many important objections,
such as lack of homogeneity of the material, effect of initial deforma-
tions, and other defects; and yet it is difficult to conceive of modern
structural design existing without the beam formula, nor is anyone
seriously disturbed because lack of homogeneity modifies somewhat
the properties of the section, or by the fact that imperfect elasticity in
the material makes invalid the superposition of stresses determined by
the beam formula for different conditions of loading. Moreover the
beam formula becomes a most inaccurate guide to the maximum
stress in any section near the point of failure; and yet it is still true
that one can scarcely conceive of modern structural design without
the guidance of the beam formula.
Similarly we say that in an elastic structure the value of E may
vary from section to section, that imperfect elasticity makes super-
position of stresses not quite correct, and that near failure the method
has only limited application. The normal process of structural design
is to determine moments and shears, and from these fiber stresses.
Whatever procedure is followed in the determination of the moments
and shears, the beam formula is used for final determination of stress.
There seem to be grounds for believing that the elastic analysis of an
arch or bent with truly fixed or truly hinged ends has greater validity
than does the method of analysis used later in design. The question
of foundation distortion and of its effect involve engineering judg-
ment. Elements involving judgment should be clearly isolated so
that the limits of such judgment can be established.
3. Acknowledgment.-The bulletin was written as a part of the
work of the Engineering Experiment Station of the University of
Illinois, of which DEAN M. S. KETCHUM is the director, and of the
Department of Civil Engineering, of which PROF. W. C. HUNTINGTON
is the head. The computations were made by M. F. LINDEMAN,
Research Graduate Assistant in Civil Engineering.
II. ANALYSES FOR FLEXURAL STRESS
4. General Equation of Flexural Stress.-Equations for stress due
to flexure are usually based on the assumption that the variation of
stress over the cross-section may be represented by a linear equation.
This assumption is based on the assumption that the beam axis is
straight and also on the assumption, based chiefly on experimental
ILLINOIS ENGINEERING EXPERIMENT STATION
observations, of the conservation of plane right sections and of the
proportionality of stress to strain.
It will be shown later that none of these assumptions is necessary,
and that the same general form of equation may be used whatever
the facts as to variation of stress intensity over the section, provided
the facts as to the shape assumed by deformed sections and the
stress-strain relations are definitely known. For the present, however,
a linear equation of stress variation over the section will be assumed.
The stress will then have the general equation f =. (a + bx + cy), in
which the coefficients a, b, c, are to be determined from the statical
conditions which state that the sum of the fiber resistances must equal
the applied load and that the sum of the moments of these fiber
resistances about any axis in the plane of the section must equal the
moment of the applied loads about that axis.
Let x, y = co5rdinates of any point on the cross-section along any
two mutually perpendicular axes X and Y through the
centroid of the section.
f = intensity of normal stress at point x, y
A = area of section
Ix = fx2dA = moment of inertia about axis Y (along the
axis X)
I, = fy2dA = moment of inertia about axis X (along the
axis Y)
Ix, = fxydA = product of inertia about axes X, Y
P = normal component of external forces
Mx = moment of external forces about axis Y
My = moment of external forces about axis X
Also write
Ixy
M'x = MX - M -y
IXV
I', = I, - I'
1 y - ly I z
THE COLUMN ANALOGY 11
All these terms are practically standard in the literature of flexure
except the "skew" terms designated by primes.
Write: f = a + bx + cy
Then, from statics, IV = 0
whence: P = ffdA = af dA + bf xdA + cf ydA
Since the axes are taken through the centroid, fxdA = 0 and
P
_fydA = 0, by definition of centroid. Then P = aA, a = -
A
The total moment about the axis of Y equals zero, whence
Mx = ffxdA = afxdA + bfx2dA + cfxydA
= bIl + clxy (1)
The total moment about the axis of X equals zero, whence
My = ffydA = afydA + bfxydA + cfy2dA
= blie + cly (2)
IX.
Multiplying (2) by , and subtracting from (1),
II,
M, - M my M'1,
b --
Y IV
Ixy
Multiplying (1) by , and subtracting from (2),
My - Mr I- M'y
IV - I .
Hence
P M', M'..
f = + 7+."
At **',^ = 6)-_ 8.e
M, 30-(106e7)- - 9-397
t = - (+18. 14)=-Ze.l,
, ,, (-/l,e~xr.67x(+/o)_j,,.s#
A2 -"';, o.= (^ 67 ^ - - ...¢
A, = -/.yo-(-/4o.s") =-g.,_a
Taki'/ngi :;"ori:n's crbol/t f-/? ker,'.
Al "A s;'. = ^+^, r , +1z0./
667
A,= o-(+Z6.,)=-=a./,
At C+375)x"167,-(+/o)
Al "8, 6a67 =+9.38
A^s= o-(9.8) =-9.38
ILLINOIS ENGINEERING EXPERIMENT STATION
A
- N
k I +
4-
N
tN
+
I
I
4-
N
N
1-
N
+
36 '
N
Vl
N
^
^
i§
t
%
THE COLUMN ANALOGY 37
E-
n
cq
z
0
0
rfi
*
ILLINOIS ENGINEERING EXPERIMENT STATION
P = -25 X5 = -125. It acts at the centroid of the triangle of
20 1
moments. For the column section A =- = 20, I -= X 20 X 202
1 12
= 667. Applying these values
P Mxx -125 (-125) X 8.33 X (±10)
A I - 20 667
-21.86 or +9.38
Plot this moment curve, and on it as a base plot the original curve of
moments.
The same procedure is shown for cases II and III.
(b) Simple Bents
Assume the rectangular bent shown in Fig. 12. Let the loads be a
vertical load of 10k on the top and a horizontal load of 6k uniformly
distributed along one side. It is desired to draw separate moment
curves for the two cases of loading.
Assume as convenient axes a vertical through the center of b and
a horizontal through the center of aa. Tabulate length L, moment
of inertia I, horizontal coordinate of centroid x and vertical coordin-
ate of centroid y for each of the members a, b, and a.
Also record the elastic loads and their centroids. For the load of
10k, treating the girder as a simple beam, we have the moment curve
66.7
shown, average moment + - ,area loaded 3, and hence
P = +33.3 X 3 = +100. The centroid is as shown, and x =
-1.7, y = +6. Whence M. = (-1.7) X (+100) = -170.0 and
M, = (+6) X (+100) = +600.
For the horizontal load draw the curve of moments for the column
36
as a cantilever. Average moment - = -12; area loaded = 6.
Hence P = (-12) X 6 = -72. Also x = -15, y = -3.
Mý = (-15) X (-72) = +1080 and M = (-3) X (-72) =+216.
Compute a, ax, a,, ax2 + ix, ay2 + i, for each member. The
1
centroidal moment of inertia (ix and iy) equals 2 a X (projection along
the axis)2. Find the totals.
18
Reducing to the centroid, x = 0, y = + 1- - +1.2. Find Ax2,
Ay2, Px, Py, and subtract.
j
THE COLUMN ANALOGY
_ Properieos or S'e'cf'ontV
Ne Z'mr eng/1 I ar V 6y sy'/
a /ZO / /.26 0 0
144
S /1.5 6- 3 +8 ÷+264 2
c 4.0 9 a +/1 + 6
d 4.1 5 . +/4. +// 8
/19. +47'6 E43
Correct to Cenroi' +2.82 /34
/£. SO59
A/7'.Csf/ /a ,Z,,
0
+128
+197
4-
p
+430
+/1-5
+734
+734
+4-?4
+/12
+14.5
FIG. 13. BENT WITH MONITOR
The intercepts of the neutral axis for the two conditions of loading
-P/A P/A
are now found as x = M/and i = M,/I. For the vertical
M,
load compute hi , +2.1, and for the horizontal load compute
M=
vi = +0.37. The neutral axes are then plotted, and on them
the original curves of moments are drawn to the scale of distance.
+3565
+22395
+7725
-2070
+5655
+- //./
ILLINOIS ENGINEERING EXPERIMENT STATION
/Rehywraede-
N/o,"m/ni t4 ,'ber'r a at Pa/n' 0
44
-24
-18
-6
+/2
÷Z4
5
-180
-/35S
-24
+/fl0
+1-39
-39
6
4X6
438
0
0530
49
Z530
/070
7
+/1Z
+/15
+75
-9.0
7448
8
0
+80
+56
+75
-625
9
0
140
3E09
140
73595
/0'
0
6
0
+90
-/62Z
+ 68
-337
0
-360
-162O
0
-302.9
// /2 /3
+90 +/5
+2/0 + 5
-/0
I f.
/4
+ 675
+840
+33/
/5'
+ 1350J
- 5040
+ 2406
-MSl«v
Ncz~'n/ a~" Po/4t 0:
0
___ + (+~7~)('-2293) (+g~gs~l(-44o) ft A'
355 /0700 2595
Fia. 14. UNSYMMETRICAL BENT
66.7
For the vertical load plot for static moments -2.1 = 31.7 t.; for the
36
horizontal loads, 0.369 - 97.3 ft.
In Fig. 13 is shown a reinforced concrete bent having a monitor.
The dead load on the roof is assumed uniform at 1000 lb. per hori-
zontal foot.
C-ooramares or
Po&/ 0- -f241
3
75
7.5
40
/00
7-f
/ z2
a /I 2
b /5 2
c /2 3
dO 30 3
e /S 2
/6
+ 8100
+ /020
+13860
0
+43600
+40 00
+14850
-'saox
---T&&?
+36.£SO
P01121'0- -24a
-faf
= +/1.7&W
*» '
THE COLUMN ANALOGY
In this case both load and structure are symmetrical, and there is
no need to compute moment of inertia about the axis of Y. The axes
are taken on the vertical center line and through the center of mem-
bers aa.
The moment areas and their centroids for the different members
have been computed separately by breaking them up into trapezoids
and parabolas.
The same procedure is followed as in the preceding problem.
The neutral axis, however, is horizontal. hA, +11.1. Inter-
P/A
cept of neutral axis yi = 3- - -3.92. The rise of the pres-
200
sure line is -11.1 = 18.00 ft.
The signs of the intercepts of the neutral axis can usually be found
by inspection, since the neutral axis lies on the side of the centroid
opposite to the load.
In Fig. 14 is shown an unsymmetrical bent subjected to vertical
loads. The tabulation of elastic properties and of elastic loads follows
the procedure already explained. The elastic moments may also be
conveniently computed as previously explained. They have actually
been computed as the sum of the moment of the elastic load acting at
the centroid of the member plus the product of shear times elastic
centroidal moment of inertia. The method is explained later as an
extension of the analogy, but presents few advantages. The trial
axes are taken as the vertical through the center of member c and the
horizontal through the center of member a.
The correction to the centroid also follows the procedure already
explained.
The correction for dissymmetry is made as follows:
Ix, (-2854\
for I, in column (6) write I,. * = (-2854) 3209 - 2530,
I x, (-2854\
for M.in column (15),write M,. - = (+25 230) 3209 -22420
Ix, /-2854\
for I, in column (9), write Ix,. = (- 2854) \13230 = 614
Ix / - 2854\
for M, in column (16), write M,- - = (+4745) =13230- -1020.
Subtract these corrections to get I',, M'x, I',, and M',.
/
42 ILLINOIS ENGINEERING EXPERIMENT STATION
s .
VWZZi
4
z
THE COLUMN ANALOGY
NN~)
N N N
NNN
>1
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
NNNNN NN vN
N'0 N Ns ^ N NN N'0 1'
^^^^'3+4~«sr 4-I
to N^ gi
K
K
K
K
NN
'0
N
N
N
N
N
'0
N
N
N
N
N-
Ns
N
N-N
'010
N-N
N'0
'ON
t0
Ns
SN Mt-; ts '
t + *
+++4+
N\N te
NNN- i
NN9NNN
++
4+4
N~+.
++
NN
N
N
N-
N
N
N
N N
NN
+ +
NN
N-N
NN
Nt
N ~
N-N
4+
NN
toN-
N
N N
* +
N
N
N
N
N
to
+
to
N
'0
to
N-
N
N
'4
N
N
N
N
to
N
~0
N
N
N
to
N
t~1
N
N-
to
N
N
'0
N
N
N
'4
N
N
N
toN N
N N
+
to
N
N
to
to
N '0
N +
+ +
N
to
'0
N
N
I +
N
-tt*
1+
k
'4
N
s^§
NN'0 <
N
N
N
N
It)
N
N
N
%<
+ I +
II I I]
n ?
I 7
^t4
'4I
KII
N
~t~N
K ~ K '~ ~N
K~t*~ KN
oil
~
N ~
N ~t t~
-~ .~ ~
K
~N ~t~tb ~
~
S'o?
44 ILLINOIS ENGINEERING EXPERIMENT STATION
The bending moment at any point, such as joint o, may now be
found as M = m, - mi, as shown.
16. Arch Analysis.-The column analogy affords an unusually
convenient means of analyzing reinforced concrete arches, because,
once understood, it furnishes a familiar order of procedure. The
actual computations are, of course, the same as presented by other
writers.
In the case of unsymmetrical arches the equation given in this
discussion seems to offer a much more convenient order of arranging
the computations than is found elsewhere. For this reason it has
seemed worth while to give in some detail the essential steps in the
analysis of an unsymmetrical arch.
(a) Unsymmetrical Arch
The arch analyzed is shown in Fig. 15. It has a span of axis of
60 feet divided into five panels of 12 feet each. The total rise is 20
feet and the difference in level between abutments is 15 feet.
The arch axis is first divided into fifteen segments of equal hori-
zontal projections.
Use as convenient trial axes horizontal and vertical lines through
the highest point of the arch axis.
Tabulate first the known properties of the arch. These are the
length of each segment L along the arch axis, the distances x and y
to its centroid, the depth of the sections at their centers.
L L
Now compute the elastic areas a, each equal to - = 12 - ; this is
for unit width of rib. From these compute the statical moments
ax and ay about the axes of x and y, the products of inertia about these
axes ax2, ay2, axy.
Now tabulate the m, values at centroids of sections for unit loads
at each of the panel points, A, B, C, D. The statically determinate
moments will be found for these loads cantilevered from the nearer
end of the arch. The statically determinate moment on any segment
between the load and the nearer abutment, then, equals the distance
from the load to the centroid of that segment. We then compute the
elastic load P = m,a, the moment of the elastic load about the axis
of Y, Px, and about the axis of X, Py. Note that for any segment
these three quantities may be written by multiplying m, by columns
(6), (7), (8), successively.
Sum the columns for elastic area, statical moments, products of
inertia, elastic loads, elastic moments.
THE COLUMN ANALOGY
Correct to the centroid. Compute =- a and y -= and the
x A A
corrections for the products of inertia V2A, y2A, xy A, and for the
elastic moments xa;P, yZP. Subtract the corrections.
Correct for dissymmetry. Write the value of "-. IunderIand
I»
of - . My under M, and write the value of ". Ixy under Iy and of
I-. M under My. Subtract the corrections.
Draw horizontal axis through the centroid. Now compute for a
load at each panel point the values of the components of the more
M't M'.
distant reaction vi = P= and h, = - and the intercept on the
P
X axis through the centroid, x\ = - A-
Avi
Rib-shortening has not been included in the computations. It
may be corrected for by computing the average intensity of com-
pression for any given condition of loading and from this the change
of span which would take place if the arch were free to contract. This
change of span may then be treated as if it were due to change of
temperature, equivalent temperature change =- fa.
Ee
The components and location of one reaction (at the more dis-
tant abutment) now being known, it is easy to draw the pressure lines.
By scaling the ordinate from any pressure line to the kern point for
any particular cross-section and multiplying by the H value for that
pressure line, we can compute conveniently the moment about that
kern point. Thus we can draw influence lines if we wish, by plotting
kern point moments at various sections for different positions of the
unit load. From the kern moments the stresses can be computed
directly from the formula f - .
For temperature changes, the horizontal change in span if the arch
were free to expand multiplied by E is Ed X 60 ft. and the relative
vertical movement of the abutments multiplied by E is Edt X 15 ft.
Omit the constant multiplier Ed for the time being, and use it later
as a multiplier for the temperature stresses. It has been shown that
/
46 ILLINOIS ENGINEERING EXPERIMENT STATION
K
*~ k~
~
AZ
C)i
<<
K
I
NJw
THE COLUMN ANALOGY
Nmi
N
Y'4 *
- _ --
--I--
44-4+^
N N N ^
NNN^
N-CNN 'o
NsNi4Zi
NNNNi S *1
3NNNNN4N-
11,44,44,
s. N NNNN
NZ NNNN
.4NN ~
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N*VN
4-4-4-4*
~14-4
-4 N4l~
N N
N
N
N
N
N N
+4
N
N
N
N
N-.
N
N
N'
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
NN ~:s
+ + +
N
N
N
N
NN N
NN N
+ + +
N
N
N
ii
II
Ni
-4-
N
-i-i
N
N
N
WI
N
N
N
N
N
%
K
N3
N
lii
:~ -~-~ 4'N
~ -4- ~
~ '-~ ~N
-4zii~5 N ~ -~ -~ ~
~ N
N ~
N Z~ ~
N
4'
IQ
F
ILLINOIS ENGINEERING EXPERIMENT STATION
a linear displacement is analogous to a moment about the axis of
displacement. Hence we have on the analogous column elastic
moments Ms = ±15 and My = F60. That the signs of these mo-
ments are opposite is determined by the fact that the change of span
is analogous to a moment about a line connecting the ends of the arch
axis. Such a moment on the analogous column produces compression
on the top and left or on the bottom and right of the section. After
the line of action of the temperature thrust has been determined, the
sign of the bending moment at any point is readily determined for a
rise or for a fall of temperature by observing that a thrust is required
to shorten the span, and a pull to lengthen it.
These moments M. = ± 15 and My = =F60 are now corrected
for dissymmetry. The H and V components of the temperature thrust
are then computed. The thrust, of course, passes through the elastic
centroid, just as the neutral axis of a beam passes through the centroid
for pure bending.
(b) Symmetrical Arch
If the arch is symmetrical, the procedure just given is shortened.
It is now necessary to consider only one-half of the arch ring. By
inspection the product of inertia is zero and there is no correction for
dissymmetry.
In Fig. 16 is shown the analysis of an arch similar to the one just
shown except that the span is 5 panels of 16 ft. = 80 ft.
17. Haunched Beams.-Because of the occasional importance of
haunched beams in continuous frames, examples of the analysis of
such beams have been included. The arrangement of the computa-
tions is the same as in the analysis of arches except that there are no
y coordinates.
Attention is called to the computation of end moments resulting
from unit rotation of one end. These values are constants needed in
certain methods of analyzing continuous frames.* In order to com-
pute these moments a unit load is applied at one end of the section of
the analogous column and the outer fiber stresses in the column are
determined.
A beam symmetrically haunched is shown in Fig. 17 and is anal-
yzed for end moments due to a uniform load and also for end mo-
ments due to a rotation at one end.
*"Continuity as a Factor in Reinforced Concrete Design," Hardy Cross, Proceedings, A. C. I.,
Vol. 25, 1929.
"Simplified Rigid Frame Design," Report of Committee, 301, Hardy Cross, Author-Chairman,
Journal, A. C. I., Dec., 1929.
"Analysis of Continuous Frames by Distributing Fixed-End Moments," Hardy Cross, Proceed-
ings. A. S. C. E., May, 1930.
THE COLUMN ANALOGY
&' of Sp"o
6I
/ 3 /./S ±1.5 /97
2 3 /.4S ±45 098
3 3 /.75 ±7.5 0£56
4 3 2.04 ±/a5 0.35
6 3 2.35 t13.5 0.23
_409
Nu/WA/f'ed 44'- 2
18.18
ox
+2.96
±440
+3.68
±3.1
0
a'x
+
4.45
/.47
19,.80
0.73
3/.S0
a vs
38.65
056
41.90
0/17
139.33
Z7e.7
+ZZZ.5 +438
+204.7 +20V a=/Z-y
+16/.8 + 94 a=, /s
+1//.0 + 40 b l ff
+43.0 + /0 E=1
+7+ 783
+ 7. g 3
4. 1 AE=D09 - 11 -
I i 1 a be ur-l =0- n/hei--/s/:'9w " t i :
1 If sule orf4 "ro1est o 117 ufB .--.
F(lan ofthe Cholmnt Aclssorgy to T /h'o of/ Ae- rotmenon
by topl/yig' ar u/n I /ctd o t t "on u /h the cnalogous column'/e77
7 o /8 Z e 8'./(+/7-i 06h87
m' =Iomew necessary to prod'ce unit rotat/n .f
M = , omen't a" &w "0'o" 'Xe "a r,'r-a';, whet, en',
or, / for at'y g've? roion/' fat end a , w/e , "'"w fI/ee'
- . 667
FIG. 17. BEAM SYMMEmICALLY HAUNCHED
In Fig. 18 a beam unsymmetrically haunched is shown and this is
analyzed for end moments due to rotations at the ends.
18. Slopes and Deflections of Beams.-
(a) Relation of the Column Analogy to Theorems of Area-Moments
The sum of the rotations, or change in slope, on the beam cor-
responds to shear on a section through the analogous column, because
the product of moment by elastic area is rotation and the sum of load
ILLINOIS ENGINEERING EXPERIMENT STATION
---309'
t - '-',-^.
SeCIOV2 Lew'ql, d e/
35 3.33 -Z.5 8/5 -0.4
S / 3.50 +10.5 ff +.4
3 / 3.83 +//11.5 / +2.4 a-ly
4 / 4./6 +/£.5 0/7 +e.1 =
6 / 4.SO +135 0./3 + .8
6 / 4.83 +145 0.// +/.6
9.00 -/0./ 603
-1./2 //
9.0 592
(a)-For Un// Rota0/on o, "o"--o/y 0 // /u loa'd on ana/ogous
co/umn t7 / "A"
ten: m, = !+ " =+0.437, momaent necessary for un'//
rotat/on at "A .
i . = / (-13.9)A+16.1 = . N E
iat 49Z i =me - re -- --eo, moe/ a oet /i2a- eind " p'
due fo restra/7k:
(b)-for lne, o/i foat'/on at ,'"- -a/i/y unif load of area/roas
roabioction ah p tt ec
daue /, resalr n ng
Ins (a), thre, for any goie ro/uataon or eonedn at' e fd 'h ":
FIa. 18. BEAM UNSYMMETRICALLY HAUNCHED
intensity times area is shear. From the geometrical relations previ-
ously explained, it follows that the statical moment of the rotations
about any section is the displacement at that section.
Slopes along the beam, then, correspond to shears on longitudinal
sections through the analogous column and deflections of the beam
correspond to bending moments on longitudinal sections through the
analogous column parallel to the line along which the deflection
is wanted.
THE COLUMN ANALOGY
The theorems of area-moments, then, are a part of the analogy.
The theorems dealing with the slopes and deflections of beams are
among the most useful and best known in the literature of structural
analysis. They deal with displacements relative either to a tangent
to the curved beam or to a chord of the curved beam.
They may be conveniently stated as follows:
(1) (a) Slope at any point measured with reference to a tangent
to the bent beam at another point may be found as area under the
M
El curve between the two points; (b) deflections at any point meas-
ured with reference to a tangent to the bent beam at another point
may be found as the statical moment about the first point of the area
M
under the -- curve between the two points.
El
(2) (a) Slope at any point measured with reference to any chord
of the bent beam may be found as shear at that point due to the area
M
under the - curve as a load on the chord acting as a beam simply
supported at its ends; (b) deflection at any point measured with
reference to any chord of the bent beam may be found as bending
M
moment at that point due to the area under the -- curve treated as
El
a load on the chord acting as a beam simply supported at its ends.
All of these theorems are merely theorems of geometry stated in
terms convenient to the structural engineer. They neglect the effect
of distortions other than rotations, and are applicable to any curved
line where the angle changes are'very small if we substitute "angle
M
changes as loads" for "area under the - curve as a load" in the
I El
theorems.
In the column analogy the angle changes are treated as loads on
the analogous column. From the geometrical relations already
pointed out it is evident that the conception involved in the column
analogy is essentially that used in finding slopes and deflections.
If we use the column analogy in the extended form explained later
in which the angle changes are represented as forces on the analogous
column equal to moment times elastic area, and the linear distortions
are represented as couples on the analogous column equal to shear
times elastic centroidal moment of inertia, we may include at once
the effect of both angular and linear distortions. This is sometimes
a useful theorem. It is applied in the example of Fig. 14.
ILLINOIS ENGINEERING EXPERIMENT STATION
of /1co2elvs on Secti'os
/o f/'? Ao//oc?/ A116
1rh - /r7f/i1el7ce //47e
ael r'/ C 9rosn 2s'e /0
/a/ ZO,7ds.
//ev7ce li1e for A/o,?0'r/
* to oCzp/es Z4i/Ž'/e
or 5eci/,'is Para//e/ to
'- /f/7zece Z/i7e for
S7Due /o Ver/ica/ Lcwcds.
Fic. 19. INFLUENCE LINES FOR SHEAR AND MOMENT IN AN ARCH
WITH FIXED ENDS
(b) Influence Lines by the Column Analogy
The column analogy may be combined with Miller-Breslau's
principle to compute influence ordinates. According to this principle,
the influence ordinates equal the displacements of the load line which
would result from a unit distortion corresponding to the stress func-
tion under investigation. For an influence line for moment, then, we
would apply a unit rotation, for shear a unit displacement, and so on.
The application for crown moment in an arch is shown in Fig. 19.
A unit rotation is applied at the crown and the displacements of
points on the arch axis are determined. Vertical displacements along
the arch axis are influence ordinates for crown moment due to vertical
loads, horizontal displacements are influence ordinates for horizontal
loads, rotations of the arch axis are influence ordinates for applied
couples.
All of these may be found as shears and bending moments on sec-
tions through the analogous column w-hen the column is loaded with
a unit load at the crown, as shown in the figure.
Similarly for unit crown shear, apply to the column at the crown a
unit couple about the vertical axis of the arch; for crown thrust, apply
THE COLUMN ANALOGY
to the column at the crown a unit couple about the horizontal axis
of the arch.
This procedure has great value in sketching influence lines. For
numerical computation it is probably as rapid and convenient to com-
pute the influence ordinates by the elementary procedure of applying
unit loads at successive points on the arch axis.
19. Supports of the Analogous Column.-
(a) Types of Supports
The analogous column is supported on an elastic medium. The
intensity of resistance offered by this medium is the indeterminate
moment, resisting distortion, or the moment resulting from the
restraint. The intensity times the area is the rotation produced by
this resistance.
If there is a hinge in the beam, the rotation would be infinite if
there were a constant moment at the hinge; the elastic area of the
hinge is infinite. But it is inconvenient to represent an infinite area;
instead we may choose a small area of infinite stiffness-a rigid point
support. A hinge in the beam, then, may be represented by a rigid
point support of the column.
A roller nest is equivalent to a rocker-two hinges on a line normal
to the roller bed. Hence a roller nest or rocker may be represented
by two point supports on a line normal to the roller bed.
A free end would be produced by three hinges not in line. Hence,
it may be represented in the analogous column by three point sup-
ports not in line or by a fixed end.
It is interesting, though perhaps not very important, to note that
if the beam is unstable, the column is statically indeterminate; if the
beam is statically determinate, so also is the column; if the beam is
statically indeterminate, the column would be unstable if it were not
supported by the elastic medium.
In Fig. 20a is shown the analogous column and its load in the case
of a beam simply supported. The area-moment relation is familiar.
Figure 20b shows a three-hinged arch and its analogous column.
Note that in statically determinate structures only one stable curve
of moments is possible and that there is no pressure on the elastic
base-no indeterminate moment.
Figure 20c shows an arch having a crown hinge. In analyzing this
case the rigid support is given an infinite area. The centroid, then,
lies at the hinge and the total elastic area is infinite.
Figure 20d shows a cantilever beam. Note that the free end of the
beam is rigidly fixed in the analogous column. The moment area
53 y_
ILLINOIS ENGINEERING EXPERIMENT STATION
(c)-rArch w l-/a7ed £ns
S- hr~ee - &yem' Arch wk Center -1;ye
FIG. 20. TYPES OF SUPPORTS FOR ANALOGOUS COLUMNS
THE COLUMN ANALOGY
(d)-an/ e „ (e)-Arc/ w1117 Fixed na&,
w1117 Rolle- Ale,5
FIG. 20 (CONCLUDED). TYPES OF SUPPORTS FOR ANALOGOUS COLUMNS
relation is familiar to most students, as is also the conjugate beam
relation.
In Fig. 20e is shown an arch having a roller nest equivalent to two
hinges on a line normal to the roller bed. The elastic area is infinite
and the elastic moment of inertia about the axis of the roller bed-
the Y axis-is also infinite.
(b) Reciprocality of Hinges and Supports in Beam and Analogous Column
Just as hinges in the girder or arch may be represented by rigid
supports in the analogous column, so rigid supports of the girder may
be represented by hinges or combinations of hinges in the analogous
column, since bending moment in the column corresponds to deflec-
tion in the beam. If the restraint is in two directions, the column at
that point would contain two hinges at an angle with each other,
which is equivalent to a universal joint.
These relations do not, however, seem to be very useful, since it is
not convenient to analyze the stresses at the bases of such columns.
The analogy, however, is illustrated in Figs. 21a and 21b. In Fig. 21b
the analysis for moments in the beam is not affected by the hinge and
joint in the analogous column, while analysis for slopes and deflec-
tions in the beam is facilitated by their use.
+ 56
ILLINOIS ENGINEERING EXPERIMENT STATION
(O) -ec owa'/ r ra
Ohm sSwppc'^y/ Ra^'///ers-
FiG. 21. TYPES OF SUPPORTS FOR ANALOGOUS COLUMNS
(c) The Conjugate Beam
The term "conjugate beam" is due to Professor II. M. Wester-
gaard. In an article entitled "Deflection of Beams by the Conjugate
THE COLUMN ANALOGY
Beam Method"* he has shown a reciprocal relation between straight
loaded beams and certain imaginary beams which are subject to loads
equal in intensity to the curves of moments on the original beams.
The conjugate beam as presented by Professor Westergaard will
be seen to be the analogous column section. The conception, how-
ever, can be extended to curved as well as to straight beams.
IV. EXTENSION OF THE ANALOGY
20. Introduction.-The analogy between the computation of mo-
ments in beams, bents, and arches and the computation of fiber
stresses on a section of an eccentrically loaded column as previously
stated is simple and convenient. The analysis thus far explained,
however, either neglects the effect of linear distortions within the
structure or makes separate allowance for the effect of such distor-
tions. In this section of the bulletin it is explained that by an exten-
sion of the analogy it is possible to include the effect of these linear
distortions directly in the analysis. In doing this, however, the sim-
plicity of the picture is marred. The extension here presented is,
then, probably to be thought of as a very special tool to be used in-
frequently, if at all. The completeness of analysis furnished by this
method of treatment justifies the inclusion of the material.
21. Internal Distortions.-The beam formula may be modified as
follows: Moments may be separated into two parts, due to loads and
due to couples and the products of inertia may be separated into two
parts, due to the areas which make up the section and due to the
centroidal products of inertia of those areas. The terms in the beam
formula may then be redefined.
Write P = Spa
A = la
M, = 7pae, + SM,
My = Xpae, + ÷ZM,
I = 2;ax2 + 2i,
I, = 2ay2 + 2i,
Ix, = Zaxy + Zixy
Thus far in dealing with internal distortions only the angular
rotations which take place about the centroids of the small sections
into which the axis of the beam is divided have been considered.
*Loc. cit., page 7.
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 22. TERMS USED FOR INTERNAL DISTORTIONS
Angular rotations have been treated as loads on the column section
and an elastic body which will suffer angular distortion from moment
has been treated as a part of the area of the column section. But a
linear movement will result from two equal and opposite angular
movements about different centers, and if these rotations are treated
as loads, they will be parallel loads of opposite sense and will consti-
tute a couple. Hence any linear distortion within the body corre-
sponds to a moment load on the analogous column section.
The bending moment on any differential portion of a member due
to unit forces through the elastic centroid of the member is x and the
relative displacement of the ends along any axis Y due to this moment
is axy (see Fig. 22). The total displacement then is faxy and is the
product of inertia of the elastic areas about the elastic centroid. But
it makes no difference whether this displacement is due to flexural
or to shearing distortions. Whatever its cause, we can still treat it
as if it were laxy and call it the centroidal product of inertia about
the axes X and Y.
Just as any portion of a section may be represented for purposes
of beam analysis by six quantities, namely, two coordinates of its
centroid, its area, and its products of inertia about two centroidal
axes, so the elastic properties of an elastic body may be completely
represented by corresponding elastic quantities. And just as these
six elementary properties of an area may be deduced from six other
properties-namely, the statical moments about any three axes in
the plane and the products of inertia about any three pairs of axes in
the plane, so the elementary elastic properties of the section may be
deduced if we have three displacements due to unit moment on the
body and three displacements due to equal and opposite forces at the
ends of the body.
This makes it possible to include automatically all types of stregs
distortion in beams directly in the analysis, if this seems desirable.
THE COLUMN ANALOGY
Moreover it is possible in members containing straight segments sub-
ject to constant shear to evaluate the bending effects by separating
the effect of the total rotation and the relative displacement of the
ends due to bending without finding the centroid of the moment curve.
The elastic area of a body, as the term is used here, is the total
rotation produced in the body by opposite unit moments at its ends.
The elastic centroid is the center about which the rotation occurs.
The total displacement of opposite ends of a member along one
axis due to opposite unit terminal forces along any other axis is the
elastic product of inertia of the body about these axes. If the axes
are through the centroid, the displacement is an elastic centroidal
product of inertia. If the axes are coincident through the centroid,
the displacement is an elastic centroidal moment of inertia.
In computing the properties of a body as a whole, these centroidal
products of inertia may be treated just as they are treated in the com-
putation of the properties of beam sections. The elastic products of
inertia of the analogous column, then, may be computed as the sum
of the elastic areas times the products of their distances from the axes
under consideration, plus the centroidal products of inertia of the
elastic areas.
These centroidal products of inertia are useful also in computing
the elastic moments on the analogous column, though the procedure
has no simple analogue in beam analysis. It has been explained that
known linear displacements may be treated as moment loads about
the axis of displacement. Such displacements are produced by thrusts
and shears on the portions of the elastic body. The displacement
along any axis due to a force along any line equals the product of the
force by the product of inertia of the body about the axis of displace-
ment and the line of action of the force, as this product of inertia has
been defined.
The elastic moments on the analogous column may be taken for
each portion of the beam as equal to the moments at the elastic
centroid of that portion times the elastic area plus a couple equal to
the shear on that portion of the beam times the elastic centroidal
moment of inertia. In this way the elastic moments have been com-
puted in Fig. 14.
V. PHYSICAL CONSTANTS OF DEFORMATION FOR 1 '
STRUCTURAL MEMBERS
22. Nature of Physical Constants.-Thus far consideration of the
physical properties of the members has been restricted. No theory
ILLINOIS ENGINEERING EXPERIMENT STATION
has been propounded to predict the values of the elastic areas. If we
are dealing only with distortions produced by external loads, it does
not make any difference what are the absolute values of these elastic
areas, or moduli of distortion; it is the relative values only that are
wanted.
It seems somewhat unfortunate that the theory used in the anal-
ysis of continuous frames has come to be known as the theory of
elasticity. In its simplest form it has nothing to do with elasticity in
the ordinary sense of the word. The theory of elasticity merely states
the geometrical conditions essential to continuity in terms of the
physical properties of the structure. If these physical properties are
known for the conditions of stress which actually exist, then the
theory may be applied. It is thus possible to apply the theory of
continuity in a perfectly definite way to plastic materials, such as
concrete, taking account of the variation of the plastic distortions
with both the intensity and the duration of the stress, provided the
properties of the material are accurately and definitely known.
Thus, suppose an exact analysis of a concrete arch is desired and
that the ratio of total stress to total deformation is known as a func-
tion of both intensity and duration of stress. We first assume a
constant value of the ratio of stress to total deformation, which we
will call E, throughout the arch rib, and find all stresses. Since the
value of E is now a variable over each section, we transform the sec-
tions as explained previously. Using the transformed sections, the
stresses are again found and the process is repeated to any desired
degree of precision.
We now know accurately the stress conditions at the time of load-
ing. After an interval all values of E will be changed and we can re-
peat the process just outlined. By successive repetitions we could
trace out the complete stress history of the structure.
Whether we could ever know the physical properties of the mater-
ial with enough accuracy to take account of their variations is a ques-
tion of fact to be considered separately. Whether the structure is
sensitive enough to such variations in properties to make the variation
in results secured from any such analysis appreciable is also another
matter. Of what importance such information would be in the safe
and economical design of the structure is still another matter.
Qualitative thinking along these lines will disperse certain illusions
which seem to be current as to mysterious results from plastic flow
and from time yield of concrete. The subject will not be pursued
here, since the monograph is restricted to geometrical relations. The
THE COLUMN ANALOGY
important point just now is to clearly distinguish those facts which are
purely geometrical from those which are necessarily a subject for
laboratory equipment.*
23. Method of Determining Physical Constants.-Determination of
the elastic constants themselves involves a knowledge of the prop-
erties of the material. In general they are to be determined as fol-
lows: (a) apply a unit moment at each end of the body and determine
the magnitude of the rotation and the center of rotation (this gives
the elastic area and its centroid); (b) apply unit forces along an
axis Y at the ends of the member and through the elastic centroid and
determine the relative linear displacements of the ends along and
normal to axis Y. Similarly apply unit forces along an axis through
the elastic centroid along axis X, normal to axis Y, and determine
displacement along axis X (this gives Ix, Ixy, I,).
The elastic properties of a body as just defined-the elastic area,
elastic centroid, and elastic products of inertia-are true physical
properties for the stress condition in the body. If they are known for
each portion of the body, they can be computed for the whole. If
certain assumptions are made we can predict the properties of the
individual parts.
In the deductions which follow a constant value of E and the con-
servation in bending of plane right sections is assumed. In most
cases these assumptions are very nearly correct. But these values
cannot be predicted with absolute precision because of chance
variations in the properties of the material.
The elastic properties of a body are defined for two ends at which
alone forces are supposed to be applied to the body. For another pair
of termini another set of elastic properties would be deduced.
The elastic properties for a given pair of termini may be deter-
mined experimentally as follows:
Hold one of the ends rigidly and apply through a bracket attached
to the other end a vertical force and a horizontal force successively at
each of two points. In each case measure the vertical displacement
and the horizontal displacement of each of two points on the bracket
(see Fig. 23).
We now have sixteen quantities from which to deduce the six
quantities desired. If Hooke's Law holds for the material, only ten
of these quantities will be different and only six will be needed. Many
*For a treatment of these matters, see "Neglected Factors in the Analysis of Stresses in Concrete
Arches," by Lorenz G. Straub, presented as a thesis for the degree of Doctor of Philosophy at the
University of Illinois in 1927 and later published in part as "Plastic Flow in Concrete Arches," Proc., A.
S. C. E., Jan., 1930.
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 23. EXPERIMENTAL DETERMINATION OF PHYSICAL CONSTANTS
different combinations of measurements may be used. The six
quantities may all be deflections due to unit forces, as follows:
Vertical at A due to vertical force at A = Ix + Ax2
Vertical at A due to vertical force at B = Ix + Ax (x + xi)
Vertical at A due to horizontal force at A = Ix, + Axy
Vertical at B due to vertical force at B = Ix + A (x + xl)2
Horizontal at A due to horizontal force at A = I, + Ay2
Horizontal at B due to horizontal force at B = I, + A (y + yi)
From these the values of A, x, y, Ix, Iy, and Ixy may be deduced by
simple algebra for known values of xi, yI.*
If Hooke's Law does not hold, different sets of measured displace-
ments will not be consistent even though the measurements are made
with absolute accuracy, because the body does not have any one set of
elastic properties but has a different set for each different condition
of stress.
24. Computation of the Constants.-Only a few of many possible
illustrations of the computation of elastic constants is given here.
(a) Hinges and Roller Nests
Since forces acting on solid foundations produce no deformation,
the elastic constants for the earth are taken as zero if we assume
immovable abutments.
Eccentric forces acting on a frictionless hinge will produce no
linear movement of the hinge but will produce unlimited rotation;
hence the elastic area of a hinge is infinite, and its elastic centroidal
products of inertia are zero.
Forces acting on a roller nest and inclined to its bed will produce
no rotation, no displacement normal to the bed, and unlimited dis-
placement along the bed; hence a roller nest has zero elastic area, zero
*A procedure which is simpler algebraically is as follows: Apply unit moment and measure rotation
and horizontal and vertical displacement of one end. This gives the elastic area and locates the elastic
centroid. The elastic centroidal products of inertia may then be measured directly as displacements
along the centroidal axes due to unit loads through the centroid.
THE COLUMN ANALOGY
FIG. 24. FLEXURAL DISTORTION OF A BEAM
elastic centroidal moment of inertia about an axis normal to the bed,
infinite elastic centroidal moment of inertia about an axis lying in the
bed.
For a free end all elastic constants are infinite. The fact has no
particular significance except that the theorem which states the
column analogy is of perfectly general application to all plane beams,
whether statically determinate or not, both simply supported and
cantilevered.
(b) Flexural, Longitudinal, and Shearing Distortions in Straight Beams
The angle of flexure in a length of straight beam of constant sec-
tion subjected to a given bending moment is readily computed by
geometry if sections plane before bending remain plane after bending.
In Fig. 24 let a-a be given length L of the beam before bending and
b-b after bending. The angle change 4 equals the change in length of
any fiber d divided by its distance from the neutral axis y.
Then €4 =_ d. From the definition of E, d = fL where f is the fiber
y E
f My
stress along any fiber. But - - , where It is the moment of inertia
E I,
of the transformed section obtained by dividing the areas of the orig-
inal section by their values of E. Then q =- for straight beams.
It
Flexure of curved beams is discussed below.
PL
Longitudinal distortion is computed as -- for a force through the
.At
centroid of the transformed section.
The shearing distortion per unit of length of beam equals the
shear divided by the continued product of area, the shearing modulus
of elasticity and a factor depending on the shape of the section.
ILLINOIS ENGINEERING EXPERIMENT STATION
(c) Straight Homogeneous Beam of Uniform Section
Consider a straight segment of a homogeneous beam of uniform
section. Let its properties be represented by length = L, area = A,
moment of inertia = I, modulus of elasticity = E, radius of gyra-
tion = p.
Apply unit moment, unaccompanied by shear. Rotation is -
El
and centroid of rotations is at the mid-point.
Apply unit transverse shear, otherwise unaccompanied by mo-
ment. Transverse displacement of end
(a) due to bending = + fx2da = - L2
12 El
L
(b) due to shearing distortions = + A , where n is the factor
referred to above and G is the shearing modulus of elasticity. Nor-
mally, this displacement is unimportant. There is no longitudinal
displacement due to transverse shear.
Apply unit longitudinal force. Longitudinal displacement
L L
= + = + p2 - . No transverse displacement.
AE El
The elastic properties, then, are:
Elastic area = -
El
Elastic centroidal moment of inertia about longitudinal axis
1 L L
12 EI L + AGn
Elastic centroidal moment of inertia along longitudinal axis -=
AE
Elastic centroidal product of inertia about two axes = 0
Elastic centroid on axis at mid-point.
These six quantities completely describe, on the ordinary assump-
tions of mechanics, the elastic properties of this body for terminal
forces.
(d) Bars of Trusses
The strain of a bar in a truss produces a rotation at the moment
A L
center for that bar -. If a unit moment acts at that center, 0 - .
r AEr
This determines the elastic area.
THE COLUMN ANALOGY
AA
A B
FIG. 25. DEFORMATION CONSTANTS FOR WEB MEMBERS IN A TRUSS
This expression is sometimes inconvenient for web members
where chords are not parallel, because the moment center does not lie
near the panel in which the section for stress is passed. If the chords
are parallel, it leads to indeterminate expressions.
In these cases it is convenient to replace the distant elastic weight
by two elastic weights and an elastic moment of inertia lying in the
L
panel. Thus the true elastic area at 0 is + L-. If the areas at A
AEr2
and B are to have that at 0 as a resultant (see Fig. 25)
L bp L 1
a AEr2 p AErlr2 a
L ap L 1
Ab=- AEr2 p AErlr2 b
r rl r r2
since - and -
bp p ap p
For product of inertia about axes normal to the lower chord to be
the same as that of Ao,
Io-, = 0 = A.(ap)2 + Ab(bp)2 + I
L
AErr2 p2
ILLINOIS ENGINEERING EXPERIMENT STATION
-a'dicks to a/0 f'be,-
-Raedis to c'frO')/
f f-he tri/7s/ormfed
FIG. 26. SEGMENT OF A CURVED BEAM
Evidently no correction for the product of inertia is necessary if
either axis is parallel to the lower chord.
The elastic areas Aa and Ab may be located on either chord and
the moment of inertia taken along that chord provided ap and bp are
computed along that chord. The positive elastic weight always lies
on the side of the panel next to the moment center.
If the chords are parallel, the elastic areas become zero, the mo-
L
ment of inertia along the chord is still I = + AEr-r2 p2, but ri = r2.
This method of treating truss bars is sometimes advantageous for
computing stresses in indeterminate structures. In most cases
indeterminate trusses can be analyzed conveniently by other methods
than the column analogy.
(e). Beams Sharply Curved
Beams having a radius of curvature small compared with their
depth are common in machine parts; in structural design they occur
in thick arched dams and at the haunched junction of beams and
columns. The formula of Winkler, usually presented for the solution
of such beams, leads to certain complications when used to compute
deformations or for the analysis of such beams when they are statically
indeterminate. For this reason, a detailed explanation of the use of
the transformed section seems desirable here.
In Fig. 26 is shown a differential length of a beam having a center
of curvature at 0.
It has been shown that if the section is transformed by dividing
each differential area by its radius of curvature R we can write
P Me
fR = A + 7 y,
THE COLUMN ANALOGY
where f is the fiber stress at any distance yt from the centroid of the
transformed section
R is the radius of curvature of this fiber
P is the normal load
M, is the moment about the centroid of the transformed
section
At is the area of the transformed section
Is is the moment of inertia of the transformed section about
its centroid.
Also let
A = the area of the original section
Ro = the radius of curvature to the centroid of the original
section
Rt = the radius of curvature to the centroid of the trans-
formed section.
By definition At = f
Taking moments about 0
I dA
Rf
S dA At
R
The moment of inertia about 0 =
R R2 = RdA = AR.
Reducing this to the centroid of the transformed section
A
It = ARo - AtRt2 = ARo - At - Rt = A(Ro - Rt)
If, then, the area and centroid of the original section are known,
it is necessary to find only the area of the transformed section in order
to compute directly all quantities needed.
The angle of rotation about the centroid of the transformed
section is
yt Eyt
S- = -f X (length of differential fiber under consideration)
But f -Mtt
IR
ILLINOIS ENGINEERING EXPERIMENT STATION
R
Length of any fiber = R ds
Where ds is the length of the differential fiber along the axis of the
centroid of the transformed section.
ds
Hence - EI= R is the elastic area of the segment, or in general
A = angle of arc
EIt
If M, = 0, the section moves parallel to itself and
P P
A E(A ds = AE ds
E (AtR) AE
ds
Hence d- is the elastic centroidal moment of inertia for rib-shorten-
EA
ing correction.
It will be seen, then, that all expressions for beams of sharp curva-
ture take the same form as where the beams are straight if the axis of
the beam is taken as the axis defined by the centroids of the trans-
formed sections instead of by the centroids of the original sections.
(f) Compound Members-Bifurcated Members
The elastic properties of any elastic ring may be defined with
reference to the termini from the principles indicated. Thus, to
determine the elastic properties of the ring ABCDA, Fig. 27, with
reference to terminals A C, apply at C a unit moment, the ring being
cut at C and held at A. Now treat as a column section cut at A and
uniformly loaded along ABC and compute the shear at C and the
bending moment on vertical and horizontal sections through C. This
gives the elastic weight and its static moments about C, from which
the co6rdinates of the centroid may be computed.
Next apply at C a unit vertical force, the ring being cut at C and
held at A. Apply the column analogy and compute the bending
moments on the analogous columns on horizontal and vertical sec-
tions at C, the section being cut at A. This gives moment of inertia
for vertical axis through C and product of inertia for vertical and
horizontal axes through C. These may then be reduced to the
centroid.
Finally apply at C a unit horizontal force, the ring being cut at C
and held at A. Compute the bending moment on a horizontal section
through C on the analogous column section cut at A. This is the
THE COLUMN ANALOGY
moment of inertia for a horizontal axis through C. This may now
be reduced to the centroid.
If loads occur within the ring, the elastic load on the ring may be
determined from the shear (elastic load) and moments on vertical
and horizontal axes through C (static moments of the elastic load
about C) for the elastic column section if cut at A as above.
The elastic load and its point of application and also the elastic
properties of the ring being known, the ring may be treated as is
any other member.
This procedure is relatively simple but involves a good deal of
computation. The problem, however, is not a very simple one.
Anyone who has occasion to use the method for numerical problems
will find that it leads to comparatively simple expressions.
(g) Successive Compounding
The compounding procedure just indicated may be extended in-
definitely to include any number of branches. The most common and
important case is that of a series of continuous arches or bents. The
general procedure by this method in such a case is as follows:
(a) Apply a unit moment at the junction point of the outside
bent, find the elastic area and elastic centroid.
(b) Apply a unit vertical force at the junction and find the mo-
ment of inertia about a vertical axis through this junction and the
product of inertia about horizontal and vertical axes through this
point. Reduce these to the centroid.
(c) Apply a unit horizontal force at the junction and find the
centroidal moment of inertia of the combination for a horizontal axis
through the junction. Reduce this to the centroid.
(d) Substitute these elastic properties for those of the pier in the
next bent; proceed as above and continue to include any desired
number of bents.
(e) In a similar way the elastic loads are to be combined for one
bent after another.
(f) The reactions having been determined for the last bent of the
series (which may be the center bent, combinations having been made
from both ends) these can be resolved back successively through the
series.
This method has enough value to justify reference to it, though,
even in the case of continuous arches, other methods are more
convenient.
ILLINOIS ENGINEERING EXPERIMENT STATION
A
FIG. 27. CLOSED ELASTIC RING
VI. APPLICATIONS OF THEOREM
25. Fields of Application of Theorem.-The theorem here pre-
sented has several fields of application:
(1) In the routine analysis of symmetrical and unsymmetrical
arches and bents for loads, either gravity or inclined, for temperature,
for shrinkage, or for abutment displacement.
(2) In determining moments at the ends of beams fixed at ends
and in determining other properties of such beams, such as the end
moments corresponding to unit rotation at end, for use in connection
with various methods of analysis of continuous girders or frames.*
(a) It is possible thus to determine constants for use in the
general equations of displacements. These equations state that
terminal forces and moments on a member are the sum of those
due to known loads on the member or distortions in the member
when the ends are fixed and to any displacements or rotations of
the ends, known or to be determined. A special case of these
equations, applicable where the members are straight and of uni-
form section, is known in American literature as the equation of
slope-deflection.
Values of terminal forces in terms of the unknown terminal
displacements may be substituted in the equations of static
equilibrium of the joints. These equations may then be solved
for the terminal displacements.
When the joint displacements have been found the terminal
forces may be computed from the original equations of displace-
ment.
(b) Constants may be determined for equations which state
the existence of continuity at the joints. Of these the theorems of
*Loo. cit., page 48.
THE COLUMN ANALOGY
three moments and of four moments are special examples of
limited application.
(c) Terminal forces may be determined for known loads or
internal distortions on the assumption that the joints are not dis-
placed and then the unbalanced terminal forces may be distrib-
uted among the connecting members in proportion to their re-
sistance to end displacement. In a special case this is done by the
method of moment distribution. The constants needed may be
determined by the column analogy.
(d) In routine computations of slopes and deflections.
26. Methods of Analysis of Continuous Frames.-This bulletin does
not deal primarily with methods of analysis of continuous frames, but
only with the analysis and elastic properties of the individual mem-
bers of which the frame may be made up. Some comments on meth-
ods of analysis of such frames is needed to explain applications of the
column analogy indicated.
Methods of analysis of continuous frames may be divided into
those involving internal work of the frame, such as the method of
least work, and those involving the geometry of continuity. That all
methods are really the same will at once be realized but their relations
to each other will not be discussed here.
Of the geometrical methods of analysis we may distinguish those
in which the displacements of the joints are treated as unknowns and
those in which the terminal forces acting on the members at the joints
are treated as unknowns. The former are represented in a special case
by the method of slope-deflection; the latter are represented by the
theorem of three moments and by the theorem of four moments.
The method in which the joint displacements are treated as un-
knowns might be called the Method of Joint Displacements. In this
method the terminal shears and moments are written in terms of the
joint displacements. The equations of statics-the forces balance at
each joint-are then written for each joint. From these the terminal
displacement is computed. The terminal displacement being known,
the shears and moments may be found.
In the other method of analysis the terminal slopes and displace-
ments are stated in terms of the terminal shears and moments. The
equations of continuity which state that the displacement of a joint
is common to the ends of all members meeting at the joint are then
written for each joint. From these equations the terminal shears and
moments are found directly. The equation of three moments is a
familiar illustration of an equation of joint continuity. A less familiar
ILLINOIS ENGINEERING EXPERIMENT STATION
illustration is the equation of four moments. This method might be
called the Method of Continuity.
It is well to distinguish the method of slope-deflection from the
equation of slope-deflection; the method, which seems to have been
restricted to straight members, consists in treating joint rotations and
displacements as unknowns in equations of static equilibrium for
the joints. The equation of slope-deflection is merely one of many
forms of the equation which relates the terminal forces to the loads
and terminal displacements of a straight beam. The method of slope-
deflection may be used without using the particular form of equation
known as the equation of slope-deflection or the equation of slope-
deflection may be used without using the method of slope-deflection.
If we write the end moments and forces in terms of loads and end
displacements, we can derive from these expressions for the end ro-
tations and displacements in terms of loads and end moments and
forces. Displacements of the ends of all members at a joint are equal
and angular rotation at the ends of all members meeting at a joint are
equal. These displacements and rotations being in terms of end mo-
ments and forces on the beams and of the physical properties of the
beams, the equations can be solved for the end moments and forces.
The Method of Terminal Force Distribution will not be discussed
here. It is more closely related to the Method of Continuity than to
the Method of Joint Displacements.
27. General Equation of Displacements and Slope-Deflection.-The
general equations of end forces at any joint, A, of a structure may be
written as follows, provided, as is usually true, the principal axes of
the members are parallel and normal to each other:
da
Ma = OaNa - kbraNa + (Aa - Ab) y + M'a
F. A - A +b +Fad.a- bdb + a,
Jo Io
Ma and Fa are total end moment and end force (thrust or
shear) in the member
(Note that in general there will be two equations of force at
any joint, one for horizontal and one for vertical forces.)
M'a = moment which would exist in the member if there were
no rotation or displacement of the joints.
F'a = force which would exist in the member if there were no
rotation or displacement of the joints.
4a, Aa are respectively rotation and displacement at the joint
considered.
THE COLUMN ANALOGY 73
kb, Ab are corresponding quantities at the other end, B, of each
member successively.
Na is the moment at joint A corresponding to a unit rotation of
this joint, the other end being fixed.
ra is the carry-over factor at A (the ratio of the moment at B
due to a unit rotation at A to the moment at A due to such
a rotation.)
Io is the elastic moment of inertia about the centroid for any
member.
d is the distance of either end from the centroidal axis of a
member.
From these general forms and the equations of static equilibrium
we may write the equations for every joint in a complex structure
such as a continuous arch series or a Vierendeel truss. These equa-
tions may then be solved simultaneously for displacements and from
these the moments, shears and thrusts may be determined.
The use of such equations, requiring a carefully selected conven-
tion of signs with resulting possibility of error from this source and
involving simultaneous solution, is to be thought of as a research tool
and rarely as a tool of design.
If there are no joint displacements, or if it is convenient to make
separate allowance for such displacements, the process is much
simplified. We then have, from 2M = 0,
a M'= a + braNa
- Na zNa
the signs depending on the convention used.
This, perhaps, is more conveniently written,
M'/a raNa
0= - + Z 0
If connecting members are treated as prismatic,
I
N. = 4 r- = V2
I
M'/a L
I I
2Z -- 2 --
L L
74 ILLINOIS ENGINEERING EXPERIMENT STATION
If, further, the values M'a are due to known rotations of the bars,
6A I
M'a - L - 6E - P = 6EK4I
L
IE
K K
'a = 23 24 K -- 20b 2ZK
which expression is convenient in finding secondary stresses in trusses.
Or this may be derived from the fundamental equation above
d
SM'0 ^4akrbNa 2(Aa - Ab)
'ka ~Na + ;Na ZKa
M'a = 0, N = 4 , r = - , Aa = -
d I 1
I -L L
3ZK # SK #b
a =- 22K 2ZK
These discussions include any combination of bars of any shape
or form provided the axes are parallel or normal to each other,, and it
is not very difficult to extend the method to include skewed axes.
The slope-deflection equation for prismatic beams,
2EI
M -= (2a + kb - 3P)
may be derived in a number of ways and follows from the first general
equation when
L2
Ao - Ab d2 4
Na = 4K, ra = --Y, = - 2d , I L L = 3K
12 1
The interest in the equations at present, however, is chiefly in the
utility of the column analogy in evaluating the constants M'a, F'a,
da, Io, Na, and V. for the members.
THE COLUMN ANALOGY 75
VII. CONCLUSION
28. Conclusion.-The paper is restricted to geometrical relations
and does not discuss applications to design. Its thesis is simply that
moments, shears, slopes, and deflections of beams due to any cause
may be computed in just the same way and by the same formulas as
are used in computing reactions on a short column eccentrically
loaded or as would be used to compute shears and bending moments
on longitudinal sections through such a column.
The column analogy is a convenient tool of mechanics, a somewhat
mechanical device for a structural engineer, but one which seems to
give desired results with a minimum of thought as to method of
procedure or sign conventions. In the study of continuous frames it
becomes auxiliary to methods for the analysis of such frames.
The most obvious application is in the analysis of single span
structures; perhaps its most important applications occur in the study
of continuous bents, arches, and beams.
RECENT PUBLICATIONS OF
THE ENGINEERING EXPERIMENT STATIONt
Bulletin No. 171. Heat Transfer in Ammonia Condensers, by Alonzo P. Kratz,
Horace J. Macintire, and Richard E. Gould. 1927. Thirty-five cents.
Bulletin No. 172. The Absorption of Sound by Materials, by Floyd R. Watson.
1927. Twenty-cents.
Bulletin No. 173. The Surface Tension of Molten Metals, by Earl E. Libman.
1927. Thirty cents.
*Circular No. 16. A Simple Method of Determining Stress in Curved Flexural
Members, by Benjamin J. Wilson and John F. Quereau. 1927. Fifteen cents.
Bulletin No. 174. The Effect of Climatic Changes upon a Multiple-Span Re-
inforced Concrete Arch Bridge, by Wilbur M. Wilson. 1927. Forty cents.
Bulletin No. 175. An Investigation of Web Stresses in Reinforced Concrete
Beams. Part II. Restrained Beams, by Frank E. Richart and Louis J. Larson.
1928. Forty-five cents.
Bulletin No. 176. A Metallographic Study of the Path of Fatigue Failure in
Copper, by Herbert F. Moore and Frank C. Howard. 1928. Twenty cents.
Bulletin No. 177. Embrittlement of Boiler Plate, by Samuel W. Parr and Fred-
erick G. Straub. 1928. None Available.
Bulletin No. 178. Tests on the Hydraulics and Pneumatics of House Plumbing.
Part II, by Harold E. Babbitt. 1928. Thirty-five cents.
Bulletin No. 179. An Investigation of Checkerbrick for Carbureters of Water-gas
Machines, by C. W. Parmelee, A. E. R. Westman, and W. H. Pfeiffer. 1928. Fifty
cents.
Bulletin No. 180. The Classification of Coal, by Samuel W. Parr. 1928. Thirty-
five cents.
Bulletin No. 181. The Thermal Expansion of Fireclay Bricks, by Albert E. R.
Westman. 1928. Twenty cents.
Bulletin No. 182. Flow of Brine in Pipes, by Richard E. Gould and Marion I.
Levy. 1928. Fifteen cents.
Circular No. 17. A Laboratory Furnace for Testing Resistance of Firebrick to
Slag Erosion, by Ralph K. Hursh and Chester E. Grigsby. 1928. Fifteen cents.
Bulletin No. 183. Tests of the Fatigue Strength of Steam Turbine Blade Shapes,
by Herbert F. Moore, Stuart W. Lyon, and Norville J. Alleman. 1928. Twenty-five
cents.
Bulletin No. 184. The Measurement of Air Quantities and Energy Losses in
Mine Entries. Part III, by Alfred C. Callen and Cloyde M. Smith. 1928. Thirty-
five cents.
Bulletin No. 185. A Study of the Failure of Concrete Under Combined Com-
pressive Stresses, by Frank E. Richart, Anton Brandtzaeg, and Rex L. Brown. 1928.
Fifty-five cents.
*Bulletin No. 186. Heat Transfer in Ammonia Condensers. Part II, by Alonzo
P. Kratz, Horace J. Macintire, and Richard E. Gould. 1928. Twenty cents.
*Bulletin No. 187. The Surface Tension of Molten Metals. Part II, by Earl E.
Libman. 1928. Fifteen cents.
*Bulletin No. 188. Investigation of Warm-air Furnaces and Heating Systems.
Part III, by Arthur C. Willard, Alonzo P. Kratz, and Vincent S. Day. 1928. Forty-
five cents.
Bulletin No. 189. Investigation of Warm-air Furnaces and Heating Systems.
Part IV, by Arthur C. Willard, Alonzo P. Kratz, and Vincent S. Day. 1929. Sixty
cents.
*Bulletin No. 190. The Failure of Plain and Spirally Reinforced Concrete in
Compression, by Frank E. Richart, Anton Brandtzaeg, and Rex L. Brown. 1929.
Forty cents.
*A limited number of copies of the bulletins starred are available for free distribution.
tCopies of the complete list of publications can be obtained without charge by addressing the
Engineering Experiment station, Urbana, Ill.
ILLINOIS ENGINEERING EXPERIMENT STATION
Bulletin No. 191. Rolling Tests of Plates, by Wilbur M. Wilson. 1929.
Thirty cents.
Bulletin No. 192. Investigation of Heating Rooms with Direct Steam Radiators
Equipped with Enclosures and Shields, by Arthur C. Willard, Alonzo P. Kratz,
Maurice K. Fahnestock, and Seichi Konzo. 1929. Forty cents.
Bulletin No. 193. An X-Ray Study of Firebrick, by Albert E. R. Westman.
1929. Fifteen cents.
*Bulletin No. 194. Tuning of Oscillating Circuits by Plate Current Variations,
by J. Tykocinski-Tykociner and Ralph W. Armstrong. 1929. Twenty-five cents.
Bulletin No. 195. The Plaster-Model Method of Determining Stresses Applied
to Curved Beams, by Fred B. Seely and Richard V. James. 1929. Twenty cents.
*Bulletin No. 196. An Investigation of the Friability of Different Coals, by Cloyde
M. Smith. 1929. Thirty cents.
*Circular No. 18. The Construction, Rehabilitation, and Maintenance of Gravel
Roads Suitable for Moderate Traffic, by Carroll C. Wiley. 1929. Thirty cents.
*Bulletin No. 197. A Study of Fatigue Cracks in Car Axles. Part II, by Herbert
F. Moore, Stuart W. Lyon, and Norville J. Alleman. 1929. Twenty cents.
*Bulletin No. 198. Results of Tests on Sewage Treatment, by Harold E. Babbitt
and Harry E. Schlenz. 1929. Fifty-five cents.
*Bulletin No. 199. The Measurement of Air Quantities and Energy Losses in
Mine Entries. Part IV, by Cloyde M. Smith. 1929. Thirty cents.
*Bulletin No. 200. Investigation of Endurance of Bond Strength of Various Clays
in Molding Sand, by Carl H. Casberg and William H. Spencer. 1929. Fifteen cents.
*Circular No. 19. Equipment for Gas-Liquid Reactions, by Donald B. Keyes.
1929. Ten cents.
Bulletin No. 201. Acid Resisting Cover Enamels for Sheet Iron, by Andrew I.
Andrews. 1929. Twenty-five cents.
Bulletin No. 202. Laboratory Tests of Reinforced Concrete Arch Ribs, by
Wilbur M. Wilson. 1929. Fifty-five cents.
*Bulletin No. 203. Dependability of the Theory of Concrete Arches, by Hardy
Cross. 1929. Twenty cents.
*Bulletin No. 204. The Hydroxylation of Double Bonds, by Sherlock Swann, Jr.
1930. Ten cents.
Bulletin No. 205. A Study of the Ikeda (Electrical Resistance) Short-Time Test
for Fatigue Strength of Metals, by Herbert F. Moore and Seichi Konzo. 1930.
Twenty cents.
Bulletin No. 206. Studies in Electrodeposition of Metals, by Donald B. Keyes
and Sherlock Swann, Jr. 1930. Ten cents.
*Bulletin No. 207. The Flow of Air Through Circular Orifices with Rounded
Approach, by Joseph A. Polson, Joseph G. Lowther, and Benjamin J. Wilson. 1930.
Thirty cents.
*Circular No. 20. An Electrical Method for the Determination of the Dew-Point
of Flue Gases, by Henry Fraser Johnstone. 1929. Fifteen cents.
Bulletin No. 208. A Study of Slip Lines, Strain Lines, and Cracks in Metals
Under Repeated Stress, by Herbert F. Moore and Tibor Ver. 1930. Thirty-five cents.
Bulletin No. 209. Heat Transfer in Ammonia Condensers. Part III, by Alonzo
P. Kratz, Horace J. Macintire, and Richard E. Gould. 1930. Thirty-five cents.
Bulletin No. 210. Tension Tests of Rivets, by Wilbur M. Wilson and William A.
Oliver. 1930. Twenty-five cents.
Bulletin No. 211. The Torsional Effect of Transverse Bending Loads on Channel
Beams, by Fred B. Seely, William J. Putnam, and William L. Schwalbe. 1930.
Thirty-five cents.
Bulletin No. 212. Stresses Due to the Pressure of One Elastic Solid Upon An-
other, by Howard R. Thomas and Victor A. Hoersch. 1930. Thirty cents.
*Bulletin No. 213. Combustion Tests With Illinois Coals, by Alonzo P. Kratz
and Wilbur J. Woodruff. 1930. Thirty cents.
*Bulletin No. 214. The Effect of Furnace Gases on the Quality of Enamels for
Sheet Steel, by Andrew I. Andrews and Emanuel A. Hertzell. 1930. Twenty cents.
*Bulletin No. 215. The Column Analogy, by Hardy Cross. 1930. Forty cents.
*A limited number of copies of the bulletins starred are available for free distribution.
UNIVERSITY OF ILLINOIS
THE STATE UNIVERSITY
URBANA
HARRY WOODBURN CHASE, Ph.D., LL.D., President
THE UNIVERSITY INCLUDES THE FOLLOWING DEPARTMENTS:
The Graduate School
The College of Liberal Arts and Sciences (Curricula: General with majors, in
the Humanities and the Sciences; Chemistry and Chemical Engineering;
Pre-legal, Pre-medical, and Pre-dental; Pre-journalism, Home Economics,
Economic Entomology, and Applied Optics)
The College of Commerce and Business Administration (Curricula: General
Business, -Banking and Finance, Insurance, Accountancy, Railway Adminis-
tration, Railway Transportation, Industrial Administration, Foreign Com-
merce, Commercial Teachers, Trade and Civic Secretarial Service, Public
Utilities, Commerce and Law)
The College of Engineering (Curricula: Architecture, Ceramics; Architectural,
Ceramic, Civil, Electrical, Gas, General, Mechanical, Mining, and Railway
Engineering; Engineering Physics)
The College of Agriculture (Curricula: General Agriculture; Floriculture; Home
Economics; Landscape Architecture; Smith-Hughes-in conjunction with
the College of Education)
The College of Education (Curricula: Two year, prescribing junior standing for
admission - General Education, Smith-Hughes Agriculture, Smith-Hughes
Home Economics, Public School Music; Four year, admitting from the high
school-Industrial Education, Athletic Coaching, Physical Education. The
SUniversity High School is the practice school of the College of Education)
The School of Music (four-year curriculum)
The College of Law (three-year curriculum based on a college degree, or three
years of college work at the University of Illinois)
The Library School (two-year curriculum for college graduates)
The School of Journalism (two-year curriculum based on two years of college
work)
The College of Medicine (in Chicago)
The College of Dentistry (in Chicago)
The School of Pharmacy (in Chicago)
The Summer Session (eight weeks)
Experiment Stations and Scientific Bureaus: U. S. Agricultural Experiment
',Station; Engineering Experiment Station; State Natural History Survey;
State Water Survey; State Geological Survey; Bureau of Educational
Research.
The Library collections contain (May 1, 1930) 836,496 volumes and 221,800
pamphlets.,
For catalogs and information address
.THE REGISTRAR'
" - Urbana, Illinola
4 44~4
5- -4
K-
45 ~ 4> - 444444
- - >4
t~sa-A~~s A <4 I 5-
4, 44 - 44 -
'44 ~44 J -4
4444 4,
>544
44-
4 4-4 444 - -- 4 44
.4' -7 / -
44 4 -4
- ~444 -5- -4-44 -
~-4 <4 1 ¾.>' -444 4>4 ~- t--.-
5 45444%,4'74 4444
44 -, 4. -4'
~ -,
> ' >44
>44
544'>, ¾ "6- 45
44< 6- 4
44 -4-4444 -
.444"
>54 44 4 4
'4 ~> -2w- 5- '4
4 54 4
* .34
~s-.- ~4t~'>-A >5->&44- 4- 4'
'-'-4 '-4
5-4 4- 544- } >44.
-4 44 4 44
-? ~~444K
4-5- X a,
~>47 5- ts.-4--.4 ~
44 -. 4 4-4 ---4
1114
~"¶ ~444 >5>44-4> <44t-457U>
44> s-.>-4 >5- .> 4$$4J444%y ~ -4k ,¶2> '~4'-X 544 - >
V.- > -4 >5 ~ ' '~ 44 .>--.r.-41 s.->>4> 4444
4; s-<4 K 54 A
4- ~*44 4'2~, 444 -444-4.»
>44444 "¶4.4
4> 4, 44-44-.> .4-44 -i~ 4->
I fl44, -4>44 ~
r~44~ - 5- X442~c-s >»">2
.' .- ' .: . ~ F . * -
¾j¶
I
-4,
S
44
'7
4~44
44
- ~ 0
4065-44
4 $
45
45
44 5
" 944
43.-4i
45- '4
44 5
'A>
>r' 4
44 $1
~44
-S
~ 2:N;st;~& .fv:~k4: <5<'> - 4-4 ~ .4