INFLUENCE CHARTS FOR COMPUTATION OF
STRESSES IN ELASTIC FOUNDATIONS
I. INTRODUCTION
1. Scope of Investigation and Summary.-This bulletin describes
a simple graphical procedure for computing stresses in the interior
of an elastic, homogeneous, isotropic solid bounded by a plane hori-
zontal surface and loaded by distributed vertical loads at the surface.
The stresses are computed from charts given herein merely by count-
ing on a chart the number of elements of area, or blocks, covered by
a plan of the loaded area drawn to proper scale and laid upon the
chart.
Influence charts of a size convenient for practical use are given in
Plates 1, 2, and 3 for computing, respectively, vertical stress on hori-
zontal planes, the sum of the principal stresses, and horizontal stress
on vertical planes, the latter two charts being constructed for a value
of Poisson's ratio of 0.5.
Influence charts are also given to a smaller scale for computing the
components of shearing stress on horizontal and vertical planes, and
for computing the corrections to the various stresses when Poisson's
ratio is different from 0.5. Numerical data and instructions are given
in the Appendix which permit drawing any of the charts to whatever
scale is desired in particular applications. For all the charts, the
influence value of the individual blocks is 0.001.
The calculation of maximum shearing stress at a point is discussed
in Chapter IV. It is shown that the so-called octahedral shear, which
in most materials is a more significant quantity than the maximum
shear, and which lies between the limits of 81.6 and 94.3 per cent of the
maximum shear, may be more easily computed than the maximum
shear, from the six components of the stresses on horizontal and
vertical planes.
The use of the influence charts is simple and rapid, and the accu-
racy of the calculations is sufficient for all practical purposes. After
a few trials one can almost guess at accurate enough values for the
stresses from a rough sketch of the loaded area.
This bulletin is not concerned with such questions as why stresses
should be computed or what should be done with the stresses after the
calculations are made. It is hoped that use of the procedure described
herein will enable time to be saved in making calculations that are at
present made by more laborious means.
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 1. SIGN CONVENTION AND NOTATION FOR AXES AND STRESSES
2. Notation and Sign Convention.-A homogeneous, elastic, iso-
tropic solid of infinite extent, bounded by a horizontal plane surface,
and loaded vertically at the surface, is considered. The x and y axes
are in the plane of the surface and the z axis is positive downward.
The notation for the stresses acting on vertical and horizontal
planes, parallel to the coordinate planes, is indicated in Fig. 1. It
will be noted that the ordinary convention for positive stresses used
in the theory of elasticity is reversed here in order that pressures or
compressions may be positive.
The intensity of load, or the load per unit of area, is denoted by p.
The stresses are denoted as follows:
p, is the normal stress on horizontal planes positive for com-
pression,
p, and p, are the normal stresses on vertical planes parallel to the
yz and the zx planes, respectively, positive for compression.
py, and p,. are horizontal shearing stresses on horizontal planes
and also vertical shearing stresses on vertical planes, positive as shown
in Fig. 1.
CHARTS FOR COMPUTATION OF STRESSES IN FOUNDATIONS
p, is the horizontal shearing stress on vertical planes, positive as
shown in Fig. 1.
The following additional notation is used:
Pvol = px + py + pz = sum of principal stresses.
y = Poisson's ratio of lateral contraction for the material.
r, a, 0 = quantities defined in Fig. 8.
A, b, C, d, Fe, = quantities defined in Equations (16) and (17).
3. Acknowledgment.-The investigation reported herein was con-
ducted as part of the work of the Engineering Experiment Station of
the University of Illinois, of which DEAN M. L. ENGER is the head.
The calculations were performed by Mr. HAROLD CRATE, student in
Civil Engineering.
A chart for computing vertical pressures, similar to that contained
herein, was described by the writer in Transactions, Am. Soc. C. E.,
Vol. 103, 1938, p. 321-324, and in Engineering News-Record, Vol. 120,
1938, p. 23-24. A condensed description of the process of constructing
such charts was also given by the writer in a paper entitled "Stress
Distribution in Soils," Proceedings of Conference on Soil Mechanics
and Its Applications, Purdue University, September, 1940, p. 295-303.
However, the material contained in this bulletin is more complete,
and the charts more convenient to use than those in the previous
publications.
II. COMPUTATION OF VERTICAL STRESS ON HORIZONTAL PLANES
4. Use of Influence Chart.-In Fig. 2 is shown a rough chart for
computing vertical stress. The chart represents a plan of the surface
of the elastic body drawn to such a scale that the length marked OQ
is the depth z at which the pressure is to be computed. The chart is
constructed of arcs of concentric circle and radial lines drawn in such
a way that each element of area bounded by two adjacent radii and
two adjacent arcs contributes the same influence to the stress.
The influence of each element of area is 0.02 times the load on the
element. Since there are 10 sectors in the diagram the influence values
of the successive circles must be 0.20, 0.40, 0.60, and 0.80. The relative
radii of the circles, in terms of the depth z from 0 to Q, are respec-
tively, 0.401, 0.637, 0.918, and 1.387. The arc for an influence of 1.0
will have an infinite radius.
A more accurate chart in which the value of each elementary area
is 0.001 is given in Plate 1. This chart is constructed in much the same
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 2. SKETCH OF INFLUENCE CHART FOR VERTICAL STRESSES
ON HORIZONTAL PLANES
manner as the chart in Fig. 2, but the areas are so chosen that they
are of convenient shape with the x and y axes both axes of symmetry.
The chart can be duplicated or drawn to other scales with the nu-
merical data given in Tables 1 and 2 in the Appendix. The manner of
using the chart is best illustrated by reference to Figs. 2 and 3.
Suppose the stress is desired at point Q' at a depth z' = 80 ft.
under point 0' in Fig. 3, where the area outlined is uniformly loaded
with a load of 5 000 lb. per sq. ft. The plan of the loaded area is re-
drawn to such a scale that the distance O'Q' in Fig. 3 becomes the
same as the distance OQ in Fig. 2. Then the drawing is placed on
Fig. 2 in such a way that point 0' falls on point 0. The number of
blocks covered by the loaded area, multiplied by the magnitude of the
load per unit of area and by the influence value of each block, gives
the stress required. In the illustration, approximately 8 blocks are
loaded, giving a stress of 0.16 times the intensity of load, or 800 lb.
per sq. ft. Plate 1 is used in the same way, except that the influence
value of each block is 0.001.
It may be noted that the stresses under similar loaded areas, at
depths proportional to the sizes of the areas, are equal. For a given
loading, different drawings of the plan of the loaded area are required
to compute stresses at various depths.
CHARTS FOR COMPUTATION OF STRESSES IN FOUNDATIONS
, '
U.
N
QI
tn.
FIG. 3. PLAN OF LOADED AREA
The chart for vertical stress on horizontal planes is radially sym-
metrical. Therefore the loading plan may be rotated through any angle
about a vertical axis through the point where the vertical stress is
computed without changing the magnitude of the stress. Consequently
the loading plan may be placed on the chart in the manner most con-
venient for the particular problem.
If the area is not uniformly loaded the chart may still be used,
provided that each influence area is considered to be loaded by the
average intensity of load on the particular block.
In using Plate 1, parts of blocks may be estimated when counting
the blocks, with sufficient accuracy for all practical purposes. In
counting the number of blocks when the number is large, the total
amount of influence within a given region may be recorded on the chart
where it can be taken into account conveniently. In general, the loaded
area will be drawn on tracing paper and laid upon the chart so that
blocks may be counted through the tracing.
III. COMPUTATION OF VARIOUS COMPONENTS OF
STRESS AT A POINT
5. General Considerations.-The stresses on any plane at a par-
ticular point can be stated in terms of the six components of stress on
an elementary cube, such as is indicated in Fig. 1. Of the six com-
ponents of stress, three are normal stresses and three are shearing
stresses. The normal and shearing stresses on any oblique plane can
be determined from a consideration of the equilibrium of an infini-
tesimal pyramid of material. The largest and smallest normal stresses
at a point occur on mutually perpendicular planes, called principal
planes, on which there is no shearing stress, and these stresses are
ILLINOIS ENGINEERING EXPERIMENT STATION
called principal stresses. There is a third principal stress of inter-
mediate magnitude on a third principal plane perpendicular to the
other principal planes, on which there is no shearing stress either. The
greatest shearing stress on any plane is found on the plane that makes
angles of 45 degrees with the directions of greatest and least principal
stresses, and that is perpendicular to the plane on which the intermedi-
ate principal stress acts. The magnitude of the maximum shearing
stress is numerically equal to one-half the algebraic difference between
the largest and the smallest principal stress.
The sum of three mutually perpendicular stresses at a point is a
constant, and is therefore equal to the sum of the principal stresses
at the point. In this bulletin this sum is denoted by the symbol pvoi
since the sum of the principal stresses is associated with the change
in volume of an elastic material.
6. Sum of Principal Stresses.-An influence chart for computing
the sum of the principal stresses, pvo,, is given in Plate 2 for a value
of Poisson's ratio of one-half. This chart is similar to that for the
computation of vertical stress on horizontal planes, and is also
radially symmetrical. Each elementary area or block on the
chart has an influence value of 0.001. To compute Pvol for other
values of Poisson's ratio than 0.5, one merely subtracts a correction
1 - 21
equal to -- times the value computed from Plate 2.
3
Since
pvol = px + Pv + Pz, (1)
and since p, can be computed from Plate 1, one can determine the
sum of p. and py, the sum of horizontal stresses on two mutually
perpendicular vertical planes, from the relation
Px + Pv = Pvoi - pN. (2)
In certain calculations this sum can be used without it being
necessary to compute p. and p, individually. This would be the.
case if one wished to compute vertical strain at a particular point,
taking into account the effect of Poisson's ratio. The vertical strain
is given by the relation
1
-W--[IP[ - A (Px + PA)]
E.
CHARTS FOR COMPUTATION OF STRESSES IN FOUNDATIONS
or, in terms of p, and pvoi,
1
-- [(1 + A) p, - pvoli] (4)
E.
where E. is the modulus of elasticity of the material.
7. Horizontal Stress on Vertical Planes.-The horizontal stress p.
on the vertical plane containing the y axis can be computed for a
value of Poisson's ratio of one-half from the influence chart given
in Plate 3. It will be noted that the elementary influence areas are
not of the same size in a given circle arc; hence, the position of the
load must be determined relative to the x and y axes. Each block
has an influence value of 0.001. Except for the fact that the chart
is not radially symmetrical, Plate 3 is used in the same way as are
Plates 1 and 2.
To determine px for a value of Poisson's ratio different from 0.5,
one subtracts two corrections from the value of p. for j = 0.5.
1 - 2A
One correction is - times the value of pvo, for 1 = 0.5 deter-
6
mined from Plate 2 and the other correction is (1 - 2,u) times the
quantity obtained from a chart shown reduced in size in Fig. 4.
For convenience in actual use such a chart should be drawn to the
same scale as Plates 1, 2, and 3. This can be done by those who
wish to do so with the numerical values of E and e given in Tables
1 and 2 of the Appendix.
In Fig. 4 each elementary block has an influence value numerically
equal to 0.001, but the sign of the influence is positive for points closer
to the x axis than to the y axis, and is negative for points closer to the
y axis than to the x axis. Thus, for a quadrant of the surface bounded
by the positive axes, the net amount of this correction is 'zero since
there is as much negative influence as positive influence.
For certain loaded areas, extending to infinity in a sector, the cor-
rection becomes infinitely large. This means that p, is infinite under
certain conditions when , is different from 0.5. This phenomenon has
been noted by Love.* Of course, this means that the material will not
remain elastic.
The same charts that are used to compute p, can also be used to
compute p1, by interchanging the x and y axes. More conveniently, p,
*A. E. H. Love, "The Stress Produced in a Semi-Infinite Solid by Pressure on Part of the
Boundary," "hil. Trans. Royal Society, London, Series A, Vol. 228, 1929, p. 377-420.
ILLINOIS ENGINEERING EXPERIMENT STATION
can be computed from the values of pvoi, Pz and p, when these are
known.
It should be noted that the vertical plane on which the stresses p,
act is part of the material and undergoes deformations. It is evident
that when loads are applied symmetrically with respect to the y axis,
there will be no deflection in the x direction of the vertical plane
through the y axis. However there will be deformations in the vertical
direction, and in the direction of y axis. Consequently, the stress act-
ing on a perfectly smooth, or frictionless, rigid vertical plane can be
computed for a given load by considering a fictitious additional load
placed in a symmetrical position. However, this merely amounts to
doubling the stress p, due to the given load.
8. Shearing Stresses on Horizontal and Vertical Planes.-Influence
charts for computing shearing stresses are shown reduced in size in
Figs. 5, 6, and 7. These can be drawn to the same scale as Plates 1, 2,
and 3 by those who will have opportunity to use them, with the data
given in Tables 1 and 2 of the Appendix.
An influence chart for computing pz, is shown in Fig. 5, constructed
from values of B and b from Tables 1 and 2. Each elementary block
has an influence value numerically equal to 0.001, but the sign of
the influence is negative when x is positive, and positive when x is
negative.
The value of p, can be computed from the same chart if the loaded
area is rotated through an angle of 90 degrees clockwise about a
vertical axis through 0.
An influence chart for computing py for a Poisson's ratio of 0.5
is shown in Fig. 6, constructed with values of F and f from Tables 1
and 2. An influence chart for computing the correction to be sub-
tracted from the value of p, obtained from Fig. 6, when Poisson's
ratio is different from 0.5, is shown in Fig. 7, constructed with values
of E and f from Tables 1 and 2. The correction is (1 - 2 jx) times the
quantity obtained from Fig. 7.
In Figs. 6 and 7, each elementary block has an influence value
numerically equal to 0.001, but the sign of the influence is positive
when x and y have the same sign, and negative when x and y are of
opposite sign.
It may be noted that Fig. 7 is identical with Fig. 4 rotated through
an angle of 45 degrees counterclockwise.
CHARTS FOR COMPUTATION OF STRESSES IN FOUNDATIONS 13
W
z
E.
z
0
0
z-
z
0
0
0
r
o
Cr)
:4.
F-.
F.
Q
(5
z
:4 .
z
0,
ri
14 ILLINOIS ENGINEERING EXPERIMENT STATION
CHARTS FOR COMPUTATION OF STRESSES IN FOUNDATIONS
16 ILLINOIS ENGINEERING EXPERIMENT STATION
z
0
z
z
z
CHARTS FOR COMPUTATION OF STRESSES IN FOUNDATIONS
IV. DETERMINATION OF MAXIMUM SHEAR
9. Principal Stresses, Maximum Shear, and Octahedral Shear.-
With the- components of stress ps, py, pz, Pzy, Pz,, and Pz,, known at a
point the stresses on any plane through the point can be determined.
Let 1, m, and n be the direction cosines, with respect to the coordinate
axes, x, y, and z, respectively, of the normal to the plane on which
stresses are to be computed. Then the normal stress p. on the plane
is as follows:*
pu = l2px 4- m2p, + n2pz + 21mpxy + 2mnpy, + 2nlp,,. (5)
The maximum shearing stress puv on the oblique plane is obtained
from pu and the resultant stress Su on the oblique plane by the
relation:
Pu2 + pu,2 = Su2 = (lp. + mp., + np,,)2
+ (lp.. + mp, + anp,)2 . (6)
+ (lp.x + mp -, + np,)2
In order that pu be a principal stress S, puv must be zero. It can be
shown that this condition requires the following relations:
IS = lp. + mpxv + np±,
mS = lpzy + mp, + npyz. (7)
nS = lp2X + mp,. + npj
To find the principal planes these equations must be solved for
1, m, and n, or rather for their ratios, since the equations are homo-
geneous. Then with the relation between the direction cosines of any
line,
S+ m + n2 = 1, (8)
one can obtain the direction cosines of the principal planes, when S is
known.
In order that Equations (7) may have a solution for 1, m, and n
*For a reference to derivation of the following equations, see, for example, S. Timoshenko,
"Theory of Elasticity," McGraw-Hill, New York, 1934, p. 182-188.
ILLINOIS ENGINEERING EXPERIMENT STATION
different from zero the determinant of the coefficients must vanish.
This condition leads to a cubic equation in S, which is:
S3 - (px + p. + p.) S2 + (ppy + PyPx + PPpx - P2xy - p2z - p2ZZ) S
- (pxpp +12p2xp± p - p.p2y,-- pp'-- Pp2,) = 0 (9)
where each of the terms in parentheses is an invariant with respect to
directions of the axes at a point.
Equation (9) will always have three real roots which will be the
three principal stresses. Let Sma and Smin denote the algebraically
largest and smallest principal stresses. Then the maximum shear rmax
is numerically equal to the quantity
1
Tmax = - (Smax - Smin). (10)
2
It will be seen that the determination of the maximum shear re-
quires the determination of the principal stresses, which is a tedious
matter involving the solution of a cubic equation.
Part of the difficulty may be avoided by computing a quantity
called the octahedral shear,* roct., which differs only slightly from
the maximum shear, and is generally a more significant quantity in
determining the behavior of materials as they approach plastic action.
The octahedral shear is defined as the shearing stress on an oblique
plane the normal to which makes equal angles with the three direc-
tions of principal stress at a point. The value of 00ct. is proportional
to the value of the so-called "shear-strain energy" or to the "energy
of distortion" of a unit volume of the material. In terms of the
principal stresses S1, S2, and S:, the value of roct. is as follows:
1 ____________(11)
Toct. = V (Sl - S2)2 + (S2 - S3)2 + 3 - )2
However, the octahedral shear can also be stated in terms of the
6 components of stress on planes parallel to the coordinate planes,
as follows:
r t.=- V (Si+S2+S)-3 (SS2 +S2S3 +S3Si) (12)
3
*See A. Nadai, "Theories of Strength," Journal of Applied Mechanics, Vol. 1, No. 3, 1933,
T. 111-129.
CHARTS FOR COMPUTATION OF STRESSES IN FOUNDATIONS
or
VT (13)
roct.--- V (p+pV+pz)2-3(pp+pp,+pz.p-p2-y -2-p2) (13)
3
Therefore, the octahedral shear can be computed from the stresses
obtained from the influence charts without first determining the
principal stresses.
It is possible to show from Equation (11) that the value of Toot.
is related to the maximum shear, Tmax, as follows:
0.8165Tmax < root. < 0.9428ma... (14)
That is, the octahedral shear always lies between the limits of 81.6
and 94.3 per cent of the maximum shear, and it is much simpler to
compute.
Illustrative Problem
To illustrate the computation of shearing stress, consider the
stresses at a depth of 20 ft. beneath the vertex of a uniformly loaded
sector of a circle of radius 100 ft., where the central angle of the
sector is 45 degrees. One side of the sector lies along the x axis, and
the other side along the line x = y. Let the load on the area be 1000
lb. per sq. ft. The value of Poisson's ratio is 0.5.
The following stresses are obtained from the formulas in the Ap-
pendix, and may be checked by use of the influence charts.
p, = 124 lb. per sq. ft.
pvoi = 301 " " " "
pX = 145 " " " "
py = 32 " " " "
p, - = -106 " " " "
py, = -44 " " " "
p, = 56 " " " "
The principal stresses are determined from the following equation
obtained by substituting the foregoing numerical values in Equa-
tion (9).
S8- 301 82 + 10 280 S-68 592 = 0.
ILLINOIS ENGINEERING EXPERIMENT STATION
The magnitude of the principal stresses can be obtained by solving
this equation by trial, or by any of the methods available for the
solution of a cubic equation. The results are, to the nearest unit,
S$ = 263 lb. per sq. ft.
S2 = 29 " " " "
S3 = 9 " " " "
Therefore the maximum shear is
rmax = -- (263 - 9) = 127 lb. per sq. ft.
The octahedral shear is determined from Equation (13) and is as
follows:
Toot. = 115 lb. per sq. ft.
APPENDIX
CONSTRUCTION OF INFLUENCE CHARTS
1. Stresses Under Vertex of Uniformly Loaded Sector of a Circle.-
Formulas have been given by Boussinesq from which one can obtain
the stresses at a point in the interior of a semi-infinite solid bounded
by a plane surface, due to a concentrated load applied normal to the
surface.* From Boussinesq's formulas one can readily obtain by inte-
gration the stresses at a depth z under the vertex of a sector of a circle
located as in Fig. 8, uniformly loaded with an intensity of load p.
With an intensity of loading of unity, the following stresses are ob-
tained at the point Q in Fig. 8:
pz = aA
p.. = -bB
pyz = - cB
1 - 2-a
pvol = pz + Pu + p, = aC - _- aC
3
1 - 2u
p, = dD - - aC - (1 - 2) eE
6
p, = fF - (1 - 2u) fE
(15)
*See, for example, N. M. Newmark, "Stress Distribution in Soils," Proceedings of Confer-
ence on Soil Mechanics and its Applications; Purdue University, September 1940, p. 295-303,
especially Equation (1).
CHARTS FOR COMPUTATION OF STRESSES IN FOUNDATIONS
where
a - --
2ir
b = sin 3
c = 1 - cos #3
2f0 + sin 20
42r
sin 20
e-
2
1 - cos 20
2
2
A = 1 - cos3 a
1
B = - sin3 a
2ir
C = 3 (1 - cos a)
3 1
D = 1 - - cos a + - cos3 a
2 2
1
E = - (log,
1
47r
1 + cos a
+ Cos a - 1)
2 cos a
3 cos a + cos3 a)
It will be noted that for ju = V2, the value
For any value of aj one has the results:
1 - 2 1
Pvol = [Pvol.-1/2- - [Pvol-1,2
3
of (1 - 2p) is zero.
1 - 2p
Px = [px]A--1 - 2- [pvoj_-i/2 - (1 - 2M) eE
6
PZY = [Ps,,.-1/2 - (1 - 24) fE
(18)
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 8. UNIFORMLY LOADED SECTOR OF A CIRCLE
2. Numerical Data for Construction of Charts.-The formulas in
the preceding section give stresses at a depth z under the vertex of a
uniformly loaded sector of a circle of radius r and central angle P/. By
combining numerical values of stresses for different sectors one can
obtain the stress at Q due to uniform load on an element of area
bounded by two radial lines and by two concentric circle arcs. The
stress at Q due to such a loading may be interpreted as an influence
coefficient. Different areas will in general produce different influences,
but values of f3 and r can be so chosen that elements of area are
defined which produce the same influence.
By proper choice of the values of a, b,... .A, B,... .F one can
find the values of P/ and a, or preferably, of P/ and tan a = r/z, which
give, when combined, influences for stresses in steps of uniform
amount. In Tables 1 and 2 values of /3 and r/z are reported, which,
when combined, correspond to elementary influence areas of value
0.001. The numerical values are so chosen that the values of #3 may
be laid out with the x and y axes always axes of symmetry for the
influence diagrams. By referring to the charts in this bulletin, the
manner of using the values of 8/ and r/z to construct influence charts
can be seen.
The chart for p, involves a and A and is independent of Ij. Since
the values of a correspond to equal divisions of a circle the chart will
CHARTS FOR COMPUTATION OF STRESSES IN FOUNDATIONS
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ILLINOIS ENGINEERING EXPERIMENT STATION
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CHARTS FOR COMPUTATION OF STRESSES IN FOUNDATIONS
be made up of circles and uniformly spaced radial lines. For con-
venience in counting it is desirable to make the elementary areas as
nearly square as possible. For this reason different numbers of radial
divisions are used in different parts of the diagram.
The same general comments apply also to pvoi which involves a and
C. The value of pvoi for / = /2 may be used also to obtain pvoi for
other values of u merely by subtracting a correction as indicated in
the previous section.
The chart for p,, involves b and B, and is independent of ju. The
elementary influence areas are negative where x is positive and positive
where x is negative. It may be observed that the same chart can be
used for Pz- and pys provided that the x and y axes are inteychanged;
or also, to prevent redrawing of the loading diagram, if the negative
y axis is taken as a new x axis and the x axis is taken as a-new y axis.
Since p, involves u it is convenient to construct a chart for p, when
IA =1 %, which involves d and D, and correction charts for the effect of
/A. One correction is proportional to pvoi as indicated by Equation
(18), and the other requires a separate chart involving e and E. In
this chart the elementary influence areas are positive when y is
numerically less than x, and negative otherwise.
The same charts can be used for p, and py with proper change of
axes, but since p. and pvoi will usually be easier to determine, py can
be obtained from the relation
PY = Pvoi - Pz - Px.
The value of p-, also depends on p, and it is expedient to construct
a chart for pa, when ju = %, which involves f and F, and a correction
chart for the effect of p.. The correction involves f and E. For both
charts the elementary influence areas are positive when x and y have
the same sign, and are negative otherwise.