Table Of ContentSTRESS ANALYSIS OF ARCH DAMS
BY
CEVDET ALI ERZEN
B.S., University of Illinois, 1941
M;S., University of Illinois, 1942
A.M., University of Illinois, 1945
THESIS
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THEORETICAL AND APPLIED MECHANICS
IN THE GRADUATE COLLEGE OF THE
UNIVERSITY OF ILLINOIS, 1951
URBANA. ILLINOIS
UNIVERSITY OF ILLINOIS
THE GRADUATE COLLEGE
January 26, 1951
T HEREBY RECOM MEXD THAT THE THESIS PkLP\RED CXDER \\\
Cevdet All Erzen
SUPERVISION BY.
Stress Analysis of Arch Sams
ENTITLED^
BE ACCEPTED IX PARTIAL FCLFILLMEXT OF THE REyLTREMEXTS FOR
Doctor of Philosophy
THE DEGREE OF In Theoretical and Applied Mechanics
£ 4,
C h.u «( ol '1 IK sis
I ll.nl III I )l)).ll ImUlt
Recommendation concurred inl
yon concurred 1117 A
Committee
on
Final Examination')-
1ir ,A
%
f Required for doctor's degree l)itt not for master's
M440
iii
TABLE OF CONTENTS
I CERTIFICATE OF APPROVAL i
II TITLE PAGE ii
III TABLE OF CONTENTS iii
iv ACia>ro";,Lrc;DG;viENT iv
V INTRODUCTION 1
VI NATURE OF PROBLEM 2
VII EXISTING I.IETHODS IN ARCK3D DAM DESIGN 4
THE CYLINDER FG- 1IULA. 4
THE TRIAL, LOAD I/ETHOD 6
VIII STRAIN EIrERGY EXPRESSION FOR CYLINDRICAL SHELLS 9
IX DEFORMATION OF SHELLS 11
X NUTAEEICAL SOLUTION OF UU^RENTTAL EQUATIONS 35
XI APPLICATION OF THE T7I7QRY 39
cm ^TJ a -v'.'p -loqr T(-,JTC! Ao
XII co: :PA .HSON OF '.RESULTS 64
GRAPHICAL COIEA iTsON OF RESULTS 65
XIII SULiL'IAHY . 66
XIV CONCLUSION 68
XV RI3LIOGAAPHY 69
XVI VITA 70"
tsaa
IV
ACiaTO"7L3DGMTCNT
The analytical study of this thesis was nade in the De
partment of Theoretical ?nd Applied hechnnics of the University
of Illinoia, of which Prof. F. ;'i. Seely is the head.
The author wishes to express his sincere appreciation to
Prof* E. L. Lan^haar .Cor his most helpful advice and assistance
in directing idiis study.
special thanks are also due to Prof. IF. 0. Idyklestad for
his oiic yur; _i.'.rujnt luri"^ th.- pr jparat:! on of this thesis.
INTRODUCTION
The design of arched dams is an important engineering problem.
Numerous methods of analysis have been proposed, but they have ei
ther been over—simplified, or they have been excessively time con
suming in practice. In this investigation a method is developed for
the stress analysis of arched dams which is based on the differential
equations of shells. The method requires some facility with mathe
matics, but it yields an answer much more quickly than the usual
method. Limits of the proportions of dams which permit the dis
carding of certain terms from the differential equations are estab
lished. Inclusion of additional terms does not alter the analysis
essentially, but it increases the computational labor. Linear varia
tion of the stress on any line normal to the middle surface has been
assumed. Test data that are presented indicate that this is a rea
sonable assumption.
The trial—load method, which is frequently used, also employs
a linear variation of stress throughout the thickness. This method
is applicable to dams of all sizes and all boundary shapes. It is
also applicable when the loads are not continuous, as in the case of
grouting of joints under pressure. The results of the trial—load
method have been well verified by tests. However, the method is
extremely tedious and lengthy, unless one has had considerable ex
perience with it.
355riEnmwffi:aamaroig
NATURE OF PROBLEM
Among the various types of dams, the most economical is the
arched dam, since the arch action utilizes the strength of the ma
terial more efficiently than the simple bending and shear action in
other types of dams. An important factor in the structural perfor
mance of an arched dam is the foundation. If the foundation is so
weak that it permits displacements of the edges of the dam, the
effectiveness of the arch action is greatly reduced. In the past,
many dams have failed because of poor foundation conditions. Now
geologists and specialists on foundations are consulted when a dam
site is selected. It is assumed in the following analysis that the
foundation holds the edges of the dam rigidly. The boundary of the
dam is assumed to be a smooth curve. Although small irregularities
ordinarily exist at the boundary, they have little effect upon the
structural performance of the dam. Large irregularities are usually
eliminated by excavation, in order to prevent severe stress con
centration.
The method that is developed in the following employs the gen
eral strain energy expression for cylindrical shells of variable
thickness. From the strain energy expression, the differential equa
tion and the boundary conditions are obtained by the principle of
virtual work, through the application of variational calculus. Since
the differential equation is so complicated that an exact solution
is impossible, a numerical procedure is employed. In this procedure,
the boundary conditions at the fixed edges are satisfied. The boun
dary conditions at the free edge and the differential equation for
the middle surface are satisfied only at isolated points. These
points are selected arbitrarily. It appears likely that an arbitra
rily close approximation can be obtained by selecting a sufficient
3
number of points. Hot\rever, the computational labor increases with
the number of points. The work oan be greatly reduced with the aid
of a net work computer.
The solving of the differential equation for the surface by
satisfying it at various points and the satisfying of the boundary
equations for the free edge necessitates a numerical solution in
the form of a series. This series is composed of terms that are
functions of the variable of the given coordinate 3ystem-, such as
w ZZ C. ,X.Y,. In order to facilitate the work, we choose the
=
functions ££ X.Y. to satisfy the forced boundary conditions. Thus,
the coefficients 0.. of the terms in this series are found by sa—
tisfying the differential equation and the boundary conditions at
isolated points.
It will be noticed that, to find these coefficients, it is
necessary to solve n + m.p simultaneous equations in which n is the
number of points treated on the surface and m represents the num
ber of boundary conditions satisfied at p points. To establish an
algebraic equation, we substitute this series in the differential
equation or the boundary equation with the known coordinates of
the point where the equation is to be satisfied. In this manner we
can set up the required number of algebraic equations for the de
termination of the coefficients in the series.
The dam treated as an illustration of the application of the
method is the Stevenson Greek Dam built in 1926 near Big Greek,
Calif., under the auspices of the Engineering Foundation. This dam
was built to study the behavior of dams under the action of water
load and varying temperature.
EXISTING METHODS IN ARCHED DAM DESIGN
The load acting on the surface of an arched dam is carried to
the foundation along the bottom and sides of the dam. The solu
tion of the problem can be simplified greatly by assuming that the
dam is made of a series of arches with no inter—action between them.
This assumption reduces the problem to one of plane stress. On this
basis, different theories have been proposed, notably, the cylinder
formula and the elastic arch theory. About twenty years ago the
trial load method was introduced with the intention of treating the
dam as a whole unit.
The elastic arch theory has received much approval and it may
be considered as a stepping stone between the cylinder formula and
the trial load method. Like the cylinder formula, the elastic arch
theory treats the structure as a series of arches. However, the
treatment of the arch under the action of water load at any eleva
tion, in the elastic arch theory is more elaborate and exact than
the simple cylinder formula.
From this discussion, it may be observed that, through the ad
vancement of the theory of arched dam design, various methods have
enjoyed wide popularity during different periods.
THE CYLINDER FORMULA METHOD
The cylinder formula is merely the formula for an infinitely
long, thin—walled cylinder under the action of constant pressure.
The stresses are assumed to be constant over any horizontal section.
This formula also ignores the interaction between arches. According
ly, the following assumptions are apparent:
1. The bending stresses in the wall are insignificant
2. The structure experiences plane stress .
5
On this basis, the compressive force acting on the cross
sectional area of an arch of unit width, with an exterior radius
r , becomes
T = Yyr
e
in which Y is "the specific weight of water and y is the distance
from the water surface to the arch section. If this expression is
divided by the thickness h, the average normal unit stress is ob
tained, ryr
e
5" ave. ~" , ~~
h h
\U*i Uj,.
In this expression the maximum unit ' \
stress differs little from the a—
verage unit stress, if h is small
compared to r . However, if the
cylinder wall is thick, the dif
ference is appreciable.
In the design of arched dams, where the allowable unit stress
is given, the thickness at any elevation is found by
ryr
a
h =
T
ave.
Using the radius to the center—line of the arch, r = r — 0.5h,
this formula is transformed into.
h 2
=
*"ave.- °'5™
Similarly, this expression in terms of the radius of the intrados
be c ome s
Yyr
h = i
^"ave. ^
The most economical layout for a given span may be estimated
on the basis of the cylinder formula. Since, an unlimited number of
arches can be laid out between two points, it is possible to select
one of these arches that renders the minimum volume of material.
This will result in a constant angle arch dam for any opening.
Denoting the central angle of the arch by 2a, the volume of a
one foot wide circular arch is given by.
V - 22 r . 2 r a
S-ave.- 0'5vy
On the other hand, r can be expressed in terms of a and the span
L of the canyon as,
rc =
2 Sina
Substituting this value of r in the expression for volume, we find
c
2(*ave.- ° -5^ Sin2«
In this, the only variables are a and y. Therefore, for any depth,
the volume may be made a minimum by taking its derivative with re
spect to a and setting this derivative equal to zero. In this manner,
we obtain, Tana = 2a . Thus, the most economical arch is the one
that has a central angle 2a = 133°—34' .
The cylinder formula is not acceptable for the analysis of
stresses in an arch dam, but it may be used satisfactorily in making
a preliminary layout of the dam. The stability of the dam can then
be studied by means of the more accurate theories.
THE TRIAL LOAD METHOD
The stress analysis of arched dams is simplified considerably
by the trial load method.
In analyzing a dam by trial load method, the first step is to
divide the structure into a series of cantilever and arch sections
