Tabulated Values for Determining the Complete
Pattern of Stresses, Strains, and Deflections
Beneath a Uniform Circular Load on a
Homogeneous Half Space
R. G. AHLVIN and H. H. ULERY, respectively. Chief and Engineer, Special Projects Section, Flexible Pavement Branch, Soils Division, U. S. Army Engineer Waterways Ejqwriment Station, Vicksburg
Results of extensive computations are presented in a manner that permits
simple determination of the complete pattern of stress, strain, and
deflection at points beneath a uniform circular load on an elastic,
homogeneous, isotropic, half space. Tabulations are presented of
eight factors for a dense network of points in terms of depth below the
load and radial distance from the load axis. Simple formulas are
presented using these factors and Poisson's ratio by which any
coordinate stress, strain, or deflection may be determined for any
value of Poisson's ratio.
•THIS PAPER presents results of computations of stress, strain, and deflection at
points beneath a uniform circular load on an elastic, homogeneous, isotropic half space.
These results are being reported in the hope that they will provide a useful reference
to anyone concerned with the distribution of stress and strain induced by tire or footing
loads that may be treated as uniformly distributed circular loads. Computations of this
type are not new {1, 2, 3); but the results presented are somewhat more extensive, provide a denser network oTvalues, give greater accuracy, and treat Poisson's ratio more
directly than previously published information. The method of presentation is also somewhat different in that tabulations have been made of each of eight functions which can be
combined through simple equations to give any coordinate stress, strain, or deflection
for any value of Poisson's ratio.
I The computations reported were made in connection with an investigation of pressures
and deflections in homogeneous soil masses. This was part of a larger study of the action
of loads on flexible pavements conducted at the U. S. Army Engineer Waterways Experinient Station. Theoretical values of stress, strain, and deflection were developed for
comparison with equivalent values measured in field test sections (7, 8, 9). These
theoretical values were computed using equations developed by Love (§) and other
methods (8 and 9).
In the computations, certain functions were obtained that repeat themselves and are
independent of Poisson's ratio. These are the functions that are tabulated and presented
vhlch combine in simple formulas to give stress, strain, or deflection for any value of
ijoisson's ratio. Tables 1 through 8 give the data in terms of depth below the loaded
area and offset distance from the center of the loaded area, of the eight functions needed
to determine any desired stress, strain, or deflection. The tables give values for
depths in radii of 0, 0.5, 1, 1. 5, 2, 2. 5, 3, 4, 5, 6, 7, 8, and 9 for offset distances
from the load axis in radii of 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, and 14. In addition,
values are presented for depths in radii of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9,
i, 1.2, 1.5, 2, 2.5, 3, and 4 at offset distances in radii of 0. 2, 0.4, 0.6, 0.8, 1.2,
aind 1.5. A few values are included for depths of 10 radii and a few for fractional radii
depths at other offset distances. The values included represent those needed for comparison with equivalent values measured in test sections as mentioned previously.
I It might appear strange that, with the modern computing equipment now available,
tabular results of the type reported are not already commonly available.
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10
The computations, however, involve elliptic functions and were made through use of
values tabulated by Legendre in 1826 (4). Direct determination of elliptic functions as
part of a computer program is sufficiently complex and time consuming that extensive
results of the type being presented herein have not been programmed, computed, and
reported.
The simple relations which combine the tabulated functions and Poisson's ratio to
give stresses, strains, or deflections are
Bulk stress
^
e = 2p(l + i.)A = (T^ +
+ CTg =
(1)
Vertical stress
(2)
= P " A + B1
Radial horizontal stress
CT = p [2vA + C + (1-2I^)f1
p
L
(3)
Tangential horizontal stress
CTg = p [2vA - D + (1 - 2I^)E]
(4)
Vertical-radial shear stress
-pz = ^zp =
(5)
Bulk strain
e =
p
^
m
(1-2.)A =
. e .
^
=
(6)
m
Vertical strain
[ ( 1 - 2I.)A + B ]
(7)
m
Radial horizontal strain
=
1+
v
[(1 - 2v)F + C ]
^
^
(8)
1+V
= P ^ ^ ^ [(1 - 2v)E - d ]
m
(9)
^m
Tangential horizontal strain
Vertical-radial shear strain
^
m
m
^
Vertical deflection
0)^ = P
1+V
r [zA + (1 - i^)H ]
(11)
m
Radial horizontal deflection
w = p - ^ (-pr) [(1 - 2 i^)E - D ] =-prfg
m
Tangential horizontal deflection
tOg = 0
(12)
(13)
11
Symbols and sign conventions follow Timoshenko (6). Figure 1 may give a clearer
concept of most of the symbols.
A, B, C . . . H
z, p, e
e
T
pz
7
e
CO
m
P
r
functions, whose values are given in Tables 1 through 8;
coordinate distances in radii, cylindrical coordinatespositive to the right, outward, and downward;
normal stress—positive for compression;
letter subscript indicates axis parallel to which the line of
action of the stress lies;
sum of three mutually perpendicular normal stresses at a point;
shear stress—positive in positive quadrant and negative in mixed
quadrant;
letter subscripts for shear stresses indicate (a) direction perpendicular to plane on which stress acts, and (b) the direction
in which it acts;
shearing strain;
subscripts for shearing strain have same meanings as those for
shear stress;
strain—positive for compression;
subscripts for strain have same meanings as those for normal
stress;
sum of three mutually perpendicular strains at a point;
deflection—positive downward and outward;
subscripts for deflection have same meanings as those for normal stress;
Poisson's ratio;
modulus of elasticity in compression—always positive;
surface contact pressure; and
radius of circular loaded area.
ELEMENTAL CUBE WITH FACES
PARALLEL TO PRINCIPAL PLANES
ELEMENTAL PRISM WITH
TWO FACES AT 4 5 ° TO
PRINCIPAL PLANES
MAJOR PRINCIPAL
PLANE
P'0,2
POINT AT WHICH
ELEMENTAL CUBES
ARE TAKEN
2'
Z
ELEMENTAL CUBE WITH FACES
PARALLEL TO AXIAL PLANES
F i g u r e 1.
Stress
directions.
V
12
From the symmetry of the circular load, It can readily be seen that there Is no shear
on vertical planes through the load axis. It follows that these must be principal planes;
therefore, the tangential-horizontal stress, a., must be a principal stress (the same Is
true for strain). The remaining two prlnclpar stresses may be obtained from the following expression:
,
^
ai, 2, s =
g
^
In which ai, a, s are principal stresses. Maximum shear stress, T^^^, may be
obtained from the expression:
''^max
_ ai - as
2
These expressions can be found elsewhere (6, 7, or in almost any textbook on mechanics).
They are Included here for ready reference."
As an example of the use of the functions and equations presented, assume that vertical stress, strain, and deflection values are desired at a depth of 3.0 radii and offset
of 6.0 radii for a Poisson's ratio of 0.3, modulus of elasticity of 10,000 psl, and surface contact pressure of 100 psl for a circular loaded area of 10-in. radius. Vertical
stress, a , will be determined from Eq. 2. Taking the value of function A from Table
1, for 3 Adii depth and 6 radii offset, to be 0. 00505 and for B from Table 2 to be
-0.00192 and substituting It In Eq. 2,
= 100 (0.00505 - 0. 001&2) = 0.313 psl
Vertical strain, t^,, is determined from Eq. 7. Inasmuch as functions A and B again
apply, the preceding values again are used. Substituting them in Eq. 7,
f ^ = 100 jp"^^^^
[(1 - 0.6)0. 00505 - 0. 00192 j = 13 x lO"' in. /in.
Vertical deflection, b^, is determined from Eq. 11. Function A again applies, but
function H must be obtained for 3 radii depth and 6 radii offset in Table 8. The value
is 0.14919. Substituting in Equation 11,
oj^ = 100
10 [(3)0. 00505 + (1 - 0.3) 0.14919] = 0. 015545 in.
In summary, this paper presents tabular values for elg^t functions that can be combined in simple equations to give values of coordinate normal or shear stress or strain
or of coordinate deflections for any desired value of Poisson's ratio for a uniformly
distributed, circular load on an elastic, homogeneous, isotropic half space. Values
are presented to five decimal place accuracy for each radius of offset from the load
axis up to 14 and for each radius of depth up to 9. Values for fractional radii offset
and depth are given to depths up to 4 radii for points beneath and just out from under the
load. The paper Is being presented for use as a reference to the computational results
included rather than for the more usual purpose of reporting recent research findings.
REFERENCES
1.
2.
3.
4.
Barber, E . S., "Application of Triaxlal Compression Test Results to the Calculation of nexlble Pavement Thickness." HRB Proc., 26 : 26-39 (1946).
Fergus, S. M . , and Miner, W. E . , "Distributed Loads on Elastic Foundations:
The Uniform Circular Load." HRB Proc., 34:582-597 (1955).
Foster, C . R . , and Ahlvin, R. G . , "Stresses and Deflections Induced by a
Uniform Circular Load." HRB Proc., 33:467-470 (1954).
Legendre, A. M . , "Traite des Foncttons EUlpUques." Vol. H, Paris (1826).
(Available at John Crerar Library, Chicago, m.)
13
5.
6.
Love, A. E . H , , "The Stress Produced in a Semi-Infinite Body by Pressure on
Part of the Boundary." Philosoph. Trans., Royal Society, Series, A, Vol. 228
(1929).
Timoshenko, S., "Theory of Elasticity." McGraw-ffill (1934).
7.
U. S. Army Engineer Waterways E:5)eriment Station, "Investigations of Pressures and Deflections for Flexible Pavements. Report 1, Homogeneous
Clayey-Silt Test Section." Tech. Memo. 3-323, Vicksburg, Miss. (March 1951).
8.
U. S. Army Engineer Waterways E^eriment Station, "Investigations of Pressures
and Deflections for Flexible Pavements. Report 3, Theoretical Stresses Induced by Uniform Circular Loads." Tech. Memo. 3-323, Vicksburg, Miss.
(Sept. 1953).
9.
U. S. Army Engineer Waterways E:q)eriment Station, "Investigations of Pressures and Deflections for Flexible Pavements. Report 4, Homogeneous Sand
Test Section." Tech. Memo. 3-323, Vicksburg, Miss. (Dec. 1954).