CONCENTRATED LOAD-CARRYING CAPACITY
OF CONCRETE SLABS ON GROUND
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By Kanakapuia S. Subba Rao 1 and Shashikant Singh 2
ABSTRACT: Lower-bound limit analysis solutions for collapse loads have been
developed for concrete slabs with the loads transmitted through circular contacts at the interior, near the edge, at the edge, or at the corner, assuming rigidplastic behavior and square yield criterion of failure. The extent of the failure
zone around the load has been determined by a statistical analysis of many
laboratory and field-test data. Existence of a ring shear at the boundary arising
from the satisfaction of conditions on moments has been established and accounted for in the analysis. Approximate solutions based on a moment reduction approach have been obtained for cases other than the interior loading case.
The collapse loads, thus obtained, are found to compare well with the available
upper-bound solutions.
INTRODUCTION
Studies on the limit state of collapse of concrete pavements a n d large
rafts subjected to concentrated loads are important for the efficient design of pavements and rafts. Meyerhof (1962) has presented limit analysis solutions for the collapse loads of plain and reinforced concrete
pavements using a square yield criterion of failure. Zingone (1972a, 1972b)
has made an upper-bound analysis of large plates on elasto-plastic soil.
Losberg (1978) developed solutions for airport pavements using the
equilibrium method of yield line theory. A similar problem was solved
using the virtual work m e t h o d by Baumann a n d Weisgerber (1983) with
the soil behavior represented b y a Winkler model.
Meyerhof's solutions are statically inadmissible, as they fail to satisfy
the shear equilibrium. Zingone's u p p e r b o u n d s , too, are inadmissible,
as the applied load and the soil reaction are not always in balance. Lowerbound limit analysis solutions for collapse loads are herein developed
for concrete slabs with the loads transmitted through circular contacts
at the interior, near the edge, at the edge, or at the corner. The analysis
is based on the rigid plastic behavior of the slab a n d the square yield
criterion of failure for concrete. A m o m e n t reduction method has been
used to develop solutions for cases where the full negative yield m o m e n t
is not developed a r o u n d the failure zone. The extent of the failure zone
around the load has been determined by a statistical analysis of m a n y
laboratory and field-test data.
COLLAPSE LOAD ANALYSIS
Interior Load
When a load m u c h less than the ultimate is applied over a small circular area on a large slab in full contact with the base, the stresses a n d
'Assoc. Prof., Dept. of Civ. Engrg., IISc, Bangalore 12, India.
2
Research Assoc, Dept. of Civ. Engrg., IISc, Bangalore 12, India.
Note.—Discussion open until May 1,1987. To extend the closing date one month,
a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 2, 1986. This paper is part of the Journal of Structural Engineering, Vol. 112,
No. 12, December, 1986. ©ASCE, ISSN 0733-9445/86/0012-2628/$01.00. Paper No.
21105.
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~iah
=S t=
FIG. 1.—Modes of Failure and Soil Pressure Distribution for Large Slab: (a) Semirigid Collapse c & a; and (b) Rigid Collapse c a a
deflections of the slab can be computed, as for an elastic and infinite
slab on an elastic subgrade (Westergaard 1948). They will all be functions
involving a characteristic length or radius of relative stiffness given by
L=
Eh3
12(1 - \i2)ks]
(1)
in which E is the modulus of elasticity of the slab; h is the thickness of
the slab; ks denotes the modulus of subgrade reaction; and |ju is the Poisson ratio of the slab (approximately 0.15).
As the load increases, the radial cracks in the slab increase until the
bending stresses along a circumferential section of the slab become equal
to the provided strength and a circumferential tension crack is formed
at the top. If the loading is flexible, like through a steel column fastened
at the base to the raft or like that of a wheel load on a pavement, a
plastic yield hinge is formed at the base of the slab and inside the loaded
area, as shown in Fig. 1(a), forming a semirigid mode of collapse. On
the other hand, if the loading is rigid, like through a column cast monolithically with the slab, a plastic yield hinge will be formed at the base
of the slab around the column, and the failure is of the rigid type [Fig.
1(b)]. A conical mode of collapse is a particular case of semirigid collapse,
with the radius of the hinge circle, c, becoming zero.
Semirigid Collapse.—Fig. 1(a) shows the collapse mode, the dimensions, and the soil pressure distribution. The load is distributed over a
radius r = a. Positive and negative circumferential yield hinges are formed
at radii r = c and r = b, respectively. The conditions on moments are
as follows:
(2a)
at r = c,Mr= +M„
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and at r=b,Mr=
-M„
(2b)
in which Mp and M„ are the positive and negative moment capacities of
the slab, respectively. The circumferential moment M e is Mv throughout
the failure region of the slab, as imposed by the chosen square yield
criterion. In addition; we have
at r = b,
dMr
=0
dr
(3)
This condition, along with the condition on M9, leads to the existence
of a boundary shear at r = b. Its magnitude can be estimated from the
moment equilibrium equation (Timoshenko and Woinowsky-Krieger 1959)
dMr
Mr + r—^-M^-rQ
dr
(4)
in which Q is the shear force per unit length. After substituting the values at r = b, the boundary shear Qb is obtained as
Qb =
'
V
(5)
b
This boundary shear has resulted on account of the conditions on moment imposed by the failure mechanism and the square yield criterion.
Losberg (1963) also points to the same. The implication of such a shear
is that the total applied load should be balanced by the distributed soil
reaction and boundary shear.
To obtain the moment field and the collapse load, Eq. 4 needs to be
integrated. The following are the salient steps:
For the zone c s r ^ a i n Fig. 1(a), the forces will be in equilibrium if
at any radius r
2itrQ = irr p - tsc' q0 - 2TT
f
q0\
fb-t\I tdt
(6)
in which p = uniform load intensity on the loaded area; q0 = maximum
soil pressure; and t varies between c and r. Similarly, for the zone a s
r<b
IttrQ =
TTH2P
- Ttc2q0 -
H^
2TT
I q0[ f
) tdt
(7)
The expressions for shear given by Eqs. 6 and 7 are substituted in Eq.
4 to get the moment equilibrium equation for each zone. Upon integration and satisfying: (1) The boundary condition given in Eq. 2; (2) radial
moment continuity condition at r = a; and (3) the boundary shear condition given by Eq. 5, the moment field is obtained as follows:
For c s r s a
Mr = Mp-P<7o
2b(r2 - 3c2) - (r 3 - 4c3) - (3c3 -- 4bc2) 12 (b - c)
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J. Struct. Eng., 1986, 112(12): 2628-2645
(8)
and for a < r <b
a'p (
1 c3\
la
J
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Mr = Mp-- -(l
r
q0c2 I
+—
c\
l - "
p
2 \
3 r 3fl2r/
2 V
rj
<7o
2
2b(r - 3c2) - (r3 - 4c3) - - (3c3 - 4k:2)
12(b - c)
(9)
The collapse load P0 is given by
P0
6TT
M„ + M„
(10)
3
2(l-p )l2a + ^
3-
1 + 2(33 - 3p 4
in which a = a/b and (3 = c/b.
Rigid Collapse.—Referring to Fig. 1(b), the hinge circle falls outside
the loaded area, with c s a. The moment field and the collapse load are
obtained as before and are given as follows:
Mr
=
a2p I
c
Mp-^(l-r
<?o
6c2(b - c) + 2b(r2 - 3c2) - (r3 - 4c3) - (2bc2 - 3c3) •
12(b - c) .
2TT (1 + 2fi3 - 3p4)
Po
and
M„ + M„
1 - 2P + 2|33 - p 4
(11)
(12)
Extent of Failure Zone
According to the elastic theory (Westergaard 1948), the maximum negative radial bending moment occurs at a distance of 1.9L from a concentrated load and occurs at increasing radii as the area of distributed
load increases. If the circumferential crack should occur at the place where
elastic theory would predict the maximum negative radial stress, the
cracking radius could be approximated for p, = 0.15 by the formula (Assur 1961):
b
a
- = 1.9 + 0.35 I-i
(13)
L-i
As the loading increases, the position of the maximum negative radial
bending moment tends to move towards the load (Augusti 1970), so that
at collapse the critical circumferential moment section with the yield hinge
circle is likely to have a radius less than that given by the elastic theory.
As applicable to plain concrete, the cracking distance for a radially cracked,
center-loaded slab (Meyerhof 1960) becomes
(14)
- = 1.63
L
VL
For a reinforced concrete slab with structurally active reinforcement,
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Losberg (1978) gives the following formula for the crack radius:
b
L
1
J
a
L
(15)
4b\
1
V1 L. yir\ \v ~^~) 3 f / J
in which y and t are soil pressure coefficients and are evaluated from
the best joining tangent line to the theoretical pressure curve. Baumann
1'
FIG. 2.—Extent of Failure Zone as Estimated by Different Theories
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and Weisgerber (1983) suggest the following:
- = 2 for interior load
L
(16a)
- = 1.5
L
(16b)
for load at edge
5
(16c)
- = 2.4 -»/ - for load at corner
L
\L
In Fig. 2 Eqs. 13-16 for crack radii are plotted. Over a hundred experimental observations made for plain, reinforced, and prestressed con-
y
•
,/
X
/
•
/
^
/
•
^
/
/
/
/
/
/
1
•
/ '
/
/
^7
•
•
/
/
• 1
/
\l1/
y
^— *•*
^
.1,
""
Conical collapse
/3 = 0 in Eq. 10
Boumann and Weisgerber
(1983)
Losborg
(1978)
//
Meyerhof
(1962)
Authors
V'
0-1
0-2
03
0-4
a/L
05
0-6
\
07
1
FIG. 3.—Collapse Loads for Large Slab with Load in Interior
2633
J. Struct. Eng., 1986, 112(12): 2628-2645
_
0-8
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crete slabs loaded at various positions are also plotted in the figure. Based
on the least square method of statistical best fit for the linear trend, the
cracking radius for the practical range of a/L, is given by
- = 0.6 + 2.3 (17)
L
L
The upper limit on the value is, however, that given by Eq. 13. The
correlation coefficient for the data is 0.75.
Collapse Loads
Substitution of Eq. 17 into Eqs. 10 and 12 will result in the collapse
loads as function of a/L. Fig. 3 presents the collapse loads according to
0-2
03
p = c/b
FIG. 4.—Collapse Load Variation with Different Modes of Collapse for Interior
Loading
2634
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different theories. Losberg's upper bounds provide the least collapse loads
up to a/L = 0.325. For higher values of a/L, Baumann and Weisgerber's
solutions provide the least values. Although Meyerhof's solutions were
obtained using the lower-bound analysis, the magnitudes of collapse load
are higher than the upper bounds. The reason can be attributed to the
noninclusion of boundary shear in the analysis and the choice of b/L.
The currently developed solution is readily seen to be very close to the
lowest solutions over the entire a/L range. The discrepancy of the current lower bound being slightly higher than the best upper bound is
due to the choice of b/L, which has been obtained empirically. Fig. 4
shows the collapse load variation with different modes of collapse. The
conical and semirigid modes of collapse show the effect of the size of
the loaded area. Once the hinge circle is outside the loaded area, the
collapse loads are only affected by the yield hinge radius and not the
radius of the loaded area.
Load Near Edge
A loading is considered to be interior if the circumferential crack at r
= b can be fully developed. Fig. 5 shows the three positions of loading
near the edge. Drucker and Hopkins (1954) showed that for a circular
plate supported along a circumference, a certain length of overhang is
necessary beyond the supports for complete mobilization of the negative
resistance. As applicable to the current problem, it can be readily shown
that the slab should extend at least up to (1 + i)b beyond the circle r =
b, if the full negative moment of Mr = -M„ = -iMv is to develop. When
the slab does not extend that far, as in the positions shown in Figs. 5(b)
and (c), a circular yield hinge at r = b will not be developed, at least
towards the edge, and the actual mode of failure would be somewhat
as shown by the dotted lines. In other words, the problem is no longer
rotationally symmetric. An approximate analysis is carried out using the
circular yield hinge at r = b, but with varying moments of resistance,
depending on how far the slab extends beyond.
Following Drucker and Hopkins (1954), it can be shown that the radial
moment that is developed at r = b is given by
-M-
4(a)
FIG. 5.—Load in Vicinity of Edge: (a) Away from Edge; (h) Near Edge; and (c)
Limiting Position of Load
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Mr = - M J 7 - 1)
(18)
R = the radial distance measured from the center. The full negative plastic moment of M„ = iMp will be developed over the region for which R
a (1 + i)b. In other words, R/b in Eq. 18 varies from 1 to (1 + i). With
reference to Fig. 5, the following relationship is written:
R = mb sec 6 with
m varying from 1 to (1 + i)
(19)
The extent of slab in any radial direction beyond r = b is then
Re = b (m sec 6 - 1)
(20)
Full negative moment will develop when Re = ib, or for the region beyond G = sec"1 (1 + i/m). If the sum of the distributed soil reaction is
s • p, in which P is the total load, then
/ a = p(i-s>
(2i)
in which Qb is the unit shear around the circumference r = b. Referring
to Eq. 5
Mr,(R\
(R\
Qb = —~\~-J; subject to the range of
l-l
from
1 to (1 + i) . . . ( 2 2 )
For a general case corresponding to Fig. 5(b)
2M„/•secp - 1 (i + A
mb
Qi> = v
+
2M,
-yr\
I
I—
o
'TT—sec" '
rir—sec
/^
I
sec 9
m
,\
( — ) (1 + O&We
(23)
When m > (1 + i), as in Fig. 5(a)
}Qb = 2(1 + iynMp
(24)
Corresponding to Fig. 5(c), in which m = 1
JQb = 2TTMP
for
i = 0
and JQb = 3.51Mp for
(25)
i=1
(26)
The collapse load analysis follows exactly the same steps as in the case
of interior loading with the substitution of Mp(R/b) for (M„ + Mp). The
collapse load for the semirigid mode is obtained as
* =
™
(27)
2(l-^)(2a + ~J
3
1 + 2p3 - 3j34
—
and for the rigid mode
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FIG. 6.—Collapse Load Variation with Different Modes of Collapse for Load near
Edge
Po = [Q](l + 2(33 - 3p4)
M 1 - 2p + 2p3 - p4
in which [Q] = $Qh/Mp. For a particular set of i and m, the boundary
shear Q6 is evaluated from Eq. 23. For m a (1 + i)/ tr*e collapse loads
will be identical with those obtained for interior loading. The collapse
loads are plotted in Fig. 6. The transformation from conical mode to rigid
mode as p increases is clearly seen in the figure.
Load at Edge
For a concentrated load applied over a small circular area at the edge
or free joint of a large slab, the stresses in the slab can be estimated
from the elastic theory (Westergaard 1948). As the load increases, the
2637
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Edge
M6 = 0
FIG. 7.—Load at Edge
14
/
/
13
Baumann and Weisgerber (1983
Losberg
_
Meyerhof
/
(1978)
(1962)
./
Authors
/
11
/
/
/ '
10
/
-
/
/
•
/
/
/
/
8
'
'
•
/
7
/
/
/
/
/
/
\
'
i
>i
1
lx
/
/
/
/
/
//
y' *
\Conical co lapse
p = 0 in Eq . 31
.6//\7'
-
•\
/
—
0-1
J
0-2
0.3
- i
0-4
a/L
•
i
0-5
0'6
0-7
FIG. 8.—Collapse Loads for Large Slab with Load at Edge
263a
J. Struct. Eng., 1986, 112(12): 2628-2645
'
0-8
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radial cracks spread outward until a circumferential crack approximating
that shown as a dotted line in Fig. 7 is formed on top of the slab extending to the free edge. The circumferential moment M„ equals Mp in
the section/g and gradually reduces to zero at the free edge. As a result,
the boundary shear Qb varies from M„/b to (M„ + Mp)/b within an angle
of TT/4 (vide Fig. 7). As a first approximation, Qb for any angle 0 measured from the free edge will be
Mp . 4
= —E I i + - I
(29)
IT
in which 9 < IT/4.
14
•
//
//
13
1
1
II
•
1
12
h
1
1
•
1 1
-
IL\
i
10
' 'I
/1
•
/
O-
+
"/I
S
•
c
/
/
//
/ /
' /
s. • _ _?_=.P_-32.S
*-*
rj.^rn
//
S
*"
A
/
0-_263_-s
®
/// //
• —
collapse
Semi- rigid
Rigid
__o.ua---'
i
Conical
collapse
collapse
/
o-p/
^
01
0-2
0-3
T«
FIG. 9.—Collapse Load Variation with Different Collapse Modes for Load at Edge
2639
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The collapse load may be estimated as follows:
1. By approximating the loaded area as a semicircular one with radius
V2 and with its center on the slab edge (Fig. 7), a collapse load corresponding to a semicircular yield hinge with Mr = -M„ at r = b can be
obtained as in a rotationally symmetric case.
2. In view of the variation of the boundry shear within the outer TT/4
sectors, the collapse load is reduced appropriately by a reduction factor.
The reduction factor to the collapse load in this case is
M,(l + 0 - - - 2 M p
RF
[H
4
v
(30)
The collapse load for the semirigid mode will become
Po
M
=
3 ( 1 + Q-7T
IT
I
(31)
2(1 - 2V2(33) 2V2a + V2—2
1 + 4V2B3 - 12B4
and for the rigid mode
Po = (1 + »>(! + 4V2"B3 - 12p4) _
M
3
1 - 2V2B + 4V2P - 4p
4
z
4
(32)
Fig. 8 contains the collapse load curves in terms of a/L and a. Solutions
of Baumann and Weisgerber (1983), Losberg (1978), and Meyerhof (1962)
are also shown in the figure. It is readily seen that the current solution
FIG. 10.—Load in Corner
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gives the least collapse load up to a/L of 0.175 and for higher values of
fl/L is close to Baumann and Weisgerber's solution, which is the least
in that region. Fig. 9 shows the collapse load variation with p, and as
affected by different modes of collapse.
Load at Corner of Quadrant
For a concentrated load applied over a small circular area adjacent to
the free corner of a large slab, as shown in Fig. 10, an approximate solution to the collapse load may be obtained, as was done for the edgeloading case. Referring to the figure, for a circular sector of central angle
X, Me changes from 0 to Mp within an angle of x/2. Qj, for any intermediate angle 9, then, is
Baumann and Weisgerber (1983 )
Losberg (Vide Meyerhol 1962)
Meyerhof
r
~7
(1962)
Authors
'
1
lj
/
/
1
-
/(
//
I
II
}
1
If
1
II
i/
u
}
-
•
/
•
/
•
/
~7
/
/
//
/
i
//
//
//
// '
//
/l
\ Conical
collapse
P =0 in Eq. 35
' f
/
^
0~ '
o'.l
'
0*
'
03
'
0-4 '
a /L
0.6
'
0-6
'
01
'
FIG. 11.—Collapse Loads for Large Slab with Load in Corner
2641
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0.8
a-
*b • ?H* •
/
_^E
.i+ b \
»»|
(33)
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The reduction factor is then
r"f/2
/•+/2
/
l?F = M p (l + 0<l»-2Mp
'Jo ( '
2
\
il/
+ - e ) d e = -M„
(34)
An approximate estimate of the collapse load may be obtained by using
an equivalent loaded area for the quadrant, with x = IT/4. The collapse
load for the semirigid mode is
22
e
_
Conical
collapse
Semi-rigid collapse
Rigid
collapse
FIG. 12.—Collapse Load Variation with Different Modes of Collapse for Load in
Corner
2642
A
J. Struct. Eng., 1986, 112(12): 2628-2645
Po =
1-5(1 + Q1T
, /
p3\
3
2(l-8p )Ua + 2 ^ j
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M
3
IT
8
1 + 16p 3 - 48p"
and for the rigid m o d e is
£o =
M
(l + i ) £ ( l + 1 6 p 3 - 4 8 p 4 )
2
1 - 4p + 16P3 - 16P4
21
8
f36N
l
'
Fig. 11 shows the collapse loads together with those presented by other
investigators. The current solution can readily be seen to be close to the
lowest upper bound. The variation of the collapse load with different
modes of failure is brought out in Fig. 12. The plots are for different
values of a, which affect the collapse load in the conical and semirigid
modes of collapse, but does not influence the rigid m o d e of collapse.
SUMMARY AND CONCLUSIONS
For an interior load o n a large concrete slab, the lower-bound limit
analysis solution is developed taking three m o d e s of collapse: conical,
semirigid, a n d rigid. During the development of a valid lower-bound
moment field for rotationally symmetric cases with the assumption of
constant circumferential yield m o m e n t , a ring shear at the outer b o u n d ary of the failure zone is imposed, and this fact is considered in the
analysis.
The radius of the negative crack or the extent of the failure zone has
been obtained from the available experimental data. The lower-bound
collapse loads thus obtained s h o w a good closeness to the best u p p e r
b o u n d s over the entire range of the loaded region.
For loadings other than in the interior, approximate solutions have
been obtained by first assuming the problem to be of rotational symmetry and then applying a correction to it in the form of a m o m e n t
reduction factor. The reduction factor is obtained b y considering the
variation of boundary shear in accordance with the slab b o u n d a r y a n d
the position of the load. The m e t h o d results in solutions that compare
well with the corresponding u p p e r b o u n d s .
APPENDIX I.—REFERENCES
Assur, A. (1961). Discussion of "Bearing capacity of floating ice sheets," by G.
G. Meyerhof. /. Engrg. Mech. DID., ASCE. 87(EM3), 63-66.
Augusti, G. (1970). "Mode approximations for rigid-plastic structures supported
by an elastic medium." International J. of Solids and Structures. 6, 809-827.
Baumann, R. A., and Weisgerber, F. E. (1983). "Yield line analysis of slabs-ongrade." /. Structural Div., ASCE. 109(ST3), 1553-1568.
Carlton, P. F., and Behrmann, R. M. (1962). Discussion of "Load carrying capacity of concrete pavements," by G. G. Meyerhof. /. Soil Mech. and Foundations
Div., ASCE. 88(SM6), 271-278.
Drucker, D. C , and Hopkins, H. G. (1954). "Combined concentrated load on
2643
J. Struct. Eng., 1986, 112(12): 2628-2645
Downloaded from ascelibrary.org by University of Wisconsin-Madison on 01/31/23. Copyright ASCE. For personal use only; all rights reserved.
ideally-plastic circular plates." Proc. Second U.S. Natl. Cong. ofAppl. Mech., 517
pp.
Gold, L. W. (1961). Discussion of "Bearing capacity of floating ice sheets," by
G. G. Meyerhof. /. Engrg. Mech. Div., ASCE. 87(EM2), 83-86.
Losberg, A. (1963). Discussion of "Load carrying capacity of concrete pavements," by G. G. Meyerhof. /. Soil Mech. and Foundations Div., ASCE. 89(SM2),
129-136.
Losberg, A. (1978). "Pavements and slabs on grade with structurally active reinforcement." /. American Concrete Inst. 75, 647-657. (
Meyerhof, G. G. (1960). "Bearing capacity of floating ice sheets." /. Engrg. Mech.
Div., ASCE. 85(EM5), 113-145.
Meyerhof, G. G. (1962). "Load carrying capacity of concrete pavements." /. Soil
Mech. and Foundations Div., ASCE. 88(SM3), 89-115.
Sebastyn, G. N., and Penner, F. (1961). Discussion of "Bearing capacity of floating ice sheets," by G. G. Meyerhof. /. Engrg. Mech. Div., ASCE. 87(EM5), 8796.
Timoshenko, S., and Woinowsky-Krieger, S. (1959). Theory of plates and shells, 2nd
ed. McGraw-Hill, New York, N.Y. 53 pp.
Westergaard, H. M. (1948). "New formulas for stresses in concrete pavements
of airfields." Trans., ASCE. 113, 425-444.
Zingone, G. (1972a). "The limit state of indefinite plates on elasto-plastic continuum." Meccanica, 7(3), 117-123.
Zingone, G. (1972b). "Probable collapse mechanisms in indefinite plates on an
elasto-plastic continuum." Meccanica. 7(3), 214-221.
APPENDIX II.—NOTATION
The following symbols are used in this paper:
a =
b =
c =
E =
h =
i =
ks =
L =
M =
Mn =
Mp =
Mr =
Me =
m =
P =
P0 =
V =
Q =
Q„ =
qo =
RF =
R,r =
s =
t =
a =
radius of loaded area;
extent of failure zone;
radius of positive yield circle;
Young's modulus of elasticity;
slab thickness;
ratio of negative to positive moment capacity;
modulus of subgrade reaction;
radius of relative stiffness;
moment capacity of slab per unit length;
negative moment capacity;
positive moment capacity;
radial moment;
circumferential moment;
factor;
load;
collapse load;
load intensity;
shear force;
boundary shear;
maximum soil pressure;
reduction factor;
radius;
factor;
soil pressure parameter;
ratio a/b;
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p = ratio c/b;
7 G =
=
(A
+ =
soil pressure parameter
angle;
Poisson ratio; and
angle.
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