I. INTRODUCTION
1. Object and Scope
The studies reported in this bulletin were undertaken in an
attempt to obtain a more rational basis for the design of the type of
building floor construction known as the two-way slab. This type
of structure consists of a reinforced concrete slab supported either
by reinforced concrete beams or by rolled steel beams arranged in a
rectangular grid pattern. The beams are in turn supported on columns
at the intersections of the grid.
The data presented herein are entirely analytical in nature and
represent two steps toward the ultimate objective of a design pro-
cedure: 1) the development of a simple, rapid, numerical procedure
for the calculation of moments in uniformly loaded continuous plates;
2) the application of this method of calculation to the study of
moments in typical two-way slab structures.
The numerical method of analysis for continuous slabs is a moment
distribution procedure, generally analogous to the Hardy Cross
method for continuous beams and frames. It provides a means for
computing the average or total moment on the edges of a uniformly
loaded rectangular plate supported on all four sides and continuous
in any directions with other rectangular plates which need not be
of the same size or shape. The average positive moments on sections
in the interior of the plate may also be obtained. Although it must
be assumed that the beams supporting the edges of the plates do
not deflect vertically, the distribution procedure may be used to
consider the torsional restraint offered by monolithically cast rein-
forced concrete beams or by monolithically cast concrete fire-proofing
encasement around steel beams.
The relative simplicity of the distribution procedure made it
possible to undertake a large number of analyses and to investigate
the effects of the following variables on the moments in two-way
slabs: (a) effects of discontinuous edges, (b) various ratios of beam
torsional stiffness to slab flexural stiffness, (c) three patterns of
loading including all panels loaded and two types of partial loading,
(d) ratios of sides of panels, and (e) combinations of panels of various
span lengths and ratios of sides.
The actual development of a design procedure, based on the
results obtained, is not presented in this bulletin. It is a separate
problem and one that requires a number of assumptions regarding
the behavior of reinforced concrete slabs. Moreover, many of these
assumptions are associated with current concepts of design and with
ILLINOIS ENGINEERING EXPERIMENT STATION
the values of moment coefficients and allowable stresses specified in
current building codes and specifications.
The results presented in this bulletin, however, have been used
in the development of a design procedure for two-way reinforced
concrete slabs, which has been described elsewhere by the writers.'
2. Outline of Bulletin
The study of moments in two-way slabs has been approached
in two distinct steps:
(a) The development of a distribution procedure for the calcula-
tions of moments in continuous plates.
(b) The use of this procedure to study the effects of several im-
portant variables on the moments in two-way slabs.
A brief outline of the remaining chapters of this bulletin is pre-
sented below:
Chapter II contains a complete description of the distribution
procedure, tables and graphs of the various constants, and an example
illustrating the use of the method.
Chapter III describes the manner in which the distribution pro-
cedure was developed and the methods used to obtain the numerical
values of the various constants. Comparisons of the results obtained
by the distribution procedure with those obtained by more exact
methods are also included.
Chapter IV contains the results of the analyses of two-way slabs
by means of the distribution procedure. Conclusions are drawn re-
garding the effects of several variables. Two types of slabs were
studied: one type consisted of twenty-five panels of equal size and
shape arranged in rows of five in each direction, while the other
consisted of sixteen panels of two different sizes and shapes.
Chapter V is a summary.
Supplementary data in the form of exact solutions for moments
in various types of slabs are presented in three appendixes.
Appendix A contains the results of a number of exact solutions
for moments in uniformly loaded rectangular plates with various edge
conditions. These moments, in all cases for a single isolated panel,
were used extensively in the derivation of the distribution constants
and in checking the distribution procedure.
Appendix B contains exact solutions for moments in uniformly
loaded continuous slabs having several panels. These solutions fur-
nished the principal basis for checking the accuracy of the distri-
bution procedure.
I C. P. Siess and N. M. Newmark. "Rational Analysis and Design of Two-Way Concrete
Slabs," Jour. American Concrete Institute, Vol. 20, No. 3, December 1948, pp. 273-315.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
Appendix C contains the results of exact solutions for moments
in rectangular plates with concentrated loads. These moments were
used to determine the applicability of the distribution procedure to
plates with concentrated loads.
The letter symbols used herein are defined where they first ap-
pear, and are assembled for convenience of reference in Appendix D.
3. Basic Assumptions
The formal distribution procedure for the analysis of continuous
slabs is based upon the exact solutions for moments in rectangular
plates presented in Appendixes A and B. These solutions were ob-
tained by means of the ordinary theory of flexure for plates, and
involve the following assumptions:
(a) The material in the slab is homogeneous, elastic, isotropic,
and of constant thickness in each panel.
(b) The resultant of the normal stress acting on any cross-section
of the slab is a pure couple.
(c) Flexural strains vary linearly through the depth of the slab.
(d) The beams exert only vertical forces on the slab; there is no
'shear between the top flange of the beam and the bottom of the slab.
(e) The reaction of the beam acts on the slab along a line and is
not distributed over a finite width. This reaction, however, may
include a moment or couple representing the torsional restraint offered
by the beam.
(f) The supporting beams do not deflect.
In all calculations in this bulletin the value of Poisson's ratio is
taken as zero. There is good reason for assuming such a value if the
results are to be applied to reinforced concrete slabs, particularly if
the conditions existing in the slab after cracking are to be considered.
For other applications of the distribution procedure, however, it may
be desirable to consider other values of Poisson's ratio. The moments
on the edges of rectangular plates supported without deflection on
all four sides are independent of the value of Poisson's ratio. The
moments in the interior of a panel of such plates are not independent
of Poisson's ratio, )u, but may be determined from the moments for
= 0 by means of the following expressions.
M1 = M0-0 + AMM-r
Mi = MA-° + AtMAo
wherein M. and M, are moments per unit of width in the direction
of x or y, respectively, and the superscripts indicate the value of
,u considered.
ILLINOIS ENGINEERING EXPERIMENT STATION
The distribution procedure, though based on the exact moments
for single panels, is nevertheless approximate. In addition to the
assumptions listed above, it was necessary in the derivation of the
distribution procedure to make additional assumptions and ap-
proximations regarding the relations between average rotations and
average moments within a panel, and between average rotations on
the common edge of adjoining panels.
None of the assumptions listed above, except possibly (c), are
satisfied completely for a reinforced concrete slab supported on
monolithically cast reinforced concrete beams. Consequently, the
moments obtained by means of the distribution procedure require
some modification before they may be used in the design of two-way
slabs. Deflection of the beams, for example, changes both the magni-
tude and the distribution of moments in the slab from those computed
by means of the distribution procedure. This problem, however, is
outside the scope of this bulletin.
4. Acknowledgment
This bulletin is based upon a thesis by C. P. Siess submitted in
partial fulfilment of the requirements for the degree of Doctor of
Philosophy in Engineering in the Graduate College of the University
of Illinois, 1948. The thesis was written under the direction of PRo-
FESSOR N. M. NEWMARK, in the Department of Civil Engineering
of which PROFESSOR W. C. HUNTINGTON is head.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
II. DISTRIBUTION PROCEDURE FOR THE COMPUTATION OF
MOMENTS IN PLATES CONTINUOUS IN Two DIRECTIONS
5. General Description of Procedure
The distribution procedure described in this chapter is applicable
to the calculation of average moments in uniformly loaded rectangular
elastic plates continuous in two directions over nondeflecting sup-
ports. The procedure is strictly analogous to the Cross' moment
distribution method for the analysis of continuous beams and frames;
that is, the procedure is one in which fixed-edge moments are calcu-
lated, unbalanced moments are distributed in proportion to the rela-
tive stiffness of the elements of the structure, and portions of the
distributed moments are "carried over" to the other edges of the
panel. The method as applied to plates, however, is not exact, since
the values of the stiffness and carry-over factors may be determined
only approximately for certain assumed conditions.
Since the moments on the edge of a continuous plate are dis-
tributed in some non-uniform manner across the width of the plate,
it is necessary to decide what value of moment is to be dealt with in
the distribution procedure. Several possibilities present themselves:
moment at the middle of the edge, maximum moment, total moment,
or average moment. From the standpoint of the physical interpre-
tation of the distribution procedure, the use of total moment is
probably to be desired. However, the relation between the shape of
the panel and the carry-over factors is somewhat simpler if the
average moment is used instead. Since the average moment is simply
the total moment divided by the width of the panel, no particular
difficulties of interpretation are introduced by this substitution.
The first step in the application of the distribution procedure is
the determination of the average moments on the edges of each
loaded panel, for all edges considered fixed. Coefficients for these
moments, as well as numerical values of the other distribution
constants, are given in the following section.
In general, the fixed-edge moments at the junction of two panels
of the slab along the line of a support will be different for the two
panels; that is, the moments at an edge will not be statically balanced.
The unbalanced moment is equal to the algebraic difference in
moments for the two panels. This unbalanced moment is distributed
to the two panels in such a manner as to equalize the moments on
the two sides of the edge, both in magnitude and in sign. If torsional
, See. for example, Hardy Cross, "Analysis of CoAntinuous Frames by Distributing Fixed-End
Moments." Trans. ASCE, Vol. 96 (1932), pp. 1-156; or Hardy Cross and Newlin I). Morgan,
"Continuous Frames of Reinforced Concrete" (John Wiley and Sons, New York, 1932), C('apter IV.
ILLINOIS ENGINEERING EXPERIMENT STATION
rigidity of the supporting beams is considered, the moments in the
beams may be handled in the same way as moments in columns of
a continuous frame. The unbalanced moments at each edge are dis-
tributed to the adjacent panels in proportion to the relative values of
the panel stiffness factors, K, as defined in the following section.
The distribution of unbalanced moments may be visualized as
the releasing of restraints on an edge which was previously held fixed.
Distribution to the various elements in proportion to their stiffness
satisfies the condition that the slopes resulting from the distributed
moment must be equal on both sides of a "joint." In this method,
it is the "average" slopes which are made equal, instead of the slopes
at every point along the edge, and therein lies one of the basic
approximations of the distribution procedure.
The redistribution of moments occurring when one edge of a
panel is released introduces additional unbalanced moments at the
other edges of that panel. These moments are said to be "carried
over" from the released edge, and their magnitude is determined
as the product of the distributed moment and certain carry-over
factors, C, which are functions of the ratio of sides, b/a, for the panel.
Four carry-over factors are needed, two for the long edges and two
for the short edges of a panel; of each pair, one determines the
moment carried over to an opposite edge, the other the moment to
an adjacent edge. Numerical values of these constants are given in
the following section.
As in the Cross method, the two operations of distributing
moments and carrying over are performed at each joint in turn, or
at all joints simultaneously, as the analyst prefers, and are then
repeated as many times as necessary until the moments on the two
sides of an edge are balanced to the degree of accuracy desired. When
this has been accomplished, the average moments on the edges of
the continuous panels may be determined by adding algebraically
the fixed-edge moments, the distributed moments, and the "carried-
over" moments at each location.
The positive moments in the interior of a panel may be obtained
as the summation of the following quantities:
(1) the moments due to the load acting on the panel simply
supported on all edges,
(2) the moments produced in the interior of the panel by the
moments acting on each edge.
The values of the moments in a simply-supported slab with uni-
form load are given in Section 6. The average interior moments
produced in each direction by the edge moments on each edge are
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
determined by means of positive moment correction factors, F, as
described in the following section. The results given by this method
of computing the positive moments in a panel are far from exact,
but it is believed that the values obtained are conservative in all
cases, without being too far on the side of safety. The principal
reason for the relative crudeness of the calculation is that the loca-
tion of the section on which the maximum positive moment occurs
varies considerably with the edge conditions of a panel. Consequently
it has been necessary to choose values of the positive moment cor-
rection factors, F, which will give a reasonably correct answer for
any condition. This subject is discussed further in Section 17.
6. Numerical Values of Constants
The numerical values of the moment coefficients and distribution
constants referred to in the preceding section are given herein. The
manner in which these values were obtained is discussed in detail
in Chapter III.
It should be pointed out that the identifying subscripts a and b,
which are used frequently in this section, refer at all times to an
edge or to a section parallel to an edge having a length of a or b,
respectively, where b is the shorter span. Thus the expression Mb
refers to the moment on the short edge of the panel, or on a section
parallel to the short edge, and not to the moment in the short span
of length b.
All the moment coefficients and distribution constants are tabu-
lated and plotted as functions of the ratio of sides, b/a. Although in
some cases values are given for the full range of b/a from zero to one,
the user is warned that the accuracy of the distribution procedure
has not been verified for b/a less than 0.5.
Fixed-Edge Moments.-The coefficients of average edge moments
for panels fixed on all sides are given in Table 1 and plotted in Fig. 1.
In both cases the values are coefficients of wb2, wherein w is the
magnitude of the uniformly distributed load per unit of area and b
is the length of the short side. These moments are exact and were
determined either directly or by interpolation from the values given
in Section 31, Appendix A.
Moments in Simply-Supported Slab.-The coefficients of average
positive moments in simply-supported slabs are given in Table 1
and in Fig. 2. As for the fixed-edge moments, the values given are
coefficients of wb2. The moments considered are the maximum in
each direction. In the short span, the maximum occurs on the sec-
tion at midspan. In the long span, the maximum occurs at midspan
ILLINOIS ENGINEERING EXPERIMENT STATION
* 00! 00
00
I I
a: a,
I =
-4 1~0 1 CC t- ~ a
0:0 00 3;? '00 0-0
Si 'I I! II I
oc^ , o3 , Cooo
co 00 c0
li
I I I I I ,
i , , I
I_ . 00 00 i S~t- 000 -K * 0000
II II I
00t00. 0000 ^ I 00-0
IF
=CC= 00co0
I I I
H
i I-~~
~%.
*74;~ ~-
~ -~ ~!~-
I - -
I --
~-- ~
.~ < ~0
0
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
for values of b, a greater than about 0.75; for smaller values the
maximum moves towards the ends of the span, and for b/a=0.5 or
less it occurs at a distance of approximately 0.33b from the short edge.
The positive moments given in Fig. 2 and Table 1 are exact values
determined from the data and sources described in Section 26,
Appendix A.
Stiffness Factors.-The stiffnesses K. and Kb represent the relation
between the average moment and the average rotation on edges a
a h
,,77mAy774y A
I
~ b
FIG. I (AT LEFT). COEFFICIENTS FOR FIXED-EDGE MOMENTS
FIG. 2 (AT RIGHT). COEFFICIENTS FOR MOMENTS IN INTERIOR OF
SIM LY-SUPPORTED SLAB
and b, respectively. Specifically, K = MAv/a,, where Mv and 4'v
refer to the same edge of a panel, for which all other edges are fixed.
The derivation of these factors is described in Sections 12 and 14
of the following chapter.
The stiffnesses for a given value of b/a are determined from the
expressions N
K, = k,, - -
K, = kb - -
b
in which N = Etl/12, the stiffness of an element of the slab, and k,, and
kb are coefficients given in Table I and lFig. 3 as functions of b/a.
ILLINOIS ENGINEERING EXPERIMENT STATION
The coefficients, k, vary with the shape of the panel in such a
manner that the stiffness factors, K, for adjacent panels having
different span lengths do not vary as much as would the corresponding
stiffnesses for a continuous beam. This is illustrated by Fig. 4, in
which the stiffnesses, K, are plotted for panels of various dimensions.
The portion of the curve on the right represents the stiffness, Kb, on
Raf/o of S/i'es, 1b/a
FIG. 3. COEFFICIENTS FOR STIFFNESS FACTORS
the short edge of a panel having a width, b, equal to unity, and a
length a varying from 1.0 to infinity. The corresponding range in Kb
is from 8.0 to 7.0. The portion of the curve on the left represents the
stiffness, Ka, on the long edge of a panel having a width a equal to
unity, and a length b varying from 1.0 to 0.4. In the range of values
from b=0.5 to 1.0, the value of K. ranges from 10.0 to 8.0, a rela-
tively small variation. For smaller values of b, however, the stiffness
approaches that for a beam and consequently varies nearly inversely
as the span length.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
Carry-Over Factors.-Consider a rectangular panel having three
edges fixed and the other simply supported. If a moment is applied
to the simply-supported edge, the moments induced on each of the
fixed edges may be expressed as the product of the applied moment
and the appropriate carry-over factor. Four such factors are required
to define the behavior of the slab. For a moment applied on a short
edge, b, the carry-over factors are Cbb for the opposite short edge,
and Cba for the adjacent long edges. Similarly, for a moment applied
on a long edge, a, the factors are C,, for the opposite long edge, and
Cub for the adjacent short edges. In each case the first letter of the
k
N
C..,
FIG. 4. VARIATION OF STIFFNESSES, K, WITH DIMENSIONS OF PANEL
subscript refers to the edge on which the moment is applied, while
the second letter refers to the edge to which the moment is "carried
over." The system of notation is further illustrated in Fig. 5.
Numerical values of the carry-over factors are given in Table 1
and Fig. 5. All the values are negative, in accordance with the sign
convention of moment adopted for use with the distribution pro-
cedure: positive moment produces compression at the top of the
slab. The carry-over factors are approximate in nature; their deriva-
tion is described in Sections 13 and 14.
Positive Moment Correction Factors.-The positive moment correc-
tion factors may be defined as the average moments produced at
certain sections in the interior of a simply-supported rectangular
panel by the application of a unit average moment to one edge of
the panel. Numerical values are given in Table 1 and Fig. 6. The
notation used is illustrated in the figure, and may be further explained
ILLINOIS ENGINEERING EXPERIMENT STATION
as follows: If a moment is applied on a short edge. b. the positive
moment correction factor applying to a section parallel to that edge
is Fbb, and the factor for a section perpendicular to that edge is Fba.
Similarly, if the moment is applied on a long edge, a, the factors are
designated F,, and F,,ab for the directions parallel and perpendicular
to that edge, respectively.
The derivation of the positive moment correction factors is
described in Section 17 of the following chapter. These constants are
-0.66
-0.56
tJ
, -0.40
, -0.36
K
-0.16
C
Ratio of Sides, b>/a
FIG. 5. VALUES OF CARRY-OVER FACTORS
only approximate and, in general, give results in somewhat poorer
agreement with the exact values than do the stiffness and carry-over
factors. For the most common conditions, the positive moments ob-
tained with the use of these factors will be quite conservative. Such
conservatism is necessary in order to take account of the fact that
the location of the maximum positive moments in a continuous panel
is dependent upon the edge conditions.
7. Modified Stiffness and Carry-Over Factors
The values given in Section 6 for the stiffness and carry-over
factors are for a panel with all edges fixed except the one to which
the constants apply. If one or more of these edges are simply sup-
ported, or if certain conditions of symmetry exist, modified values
.M0.IENTS IN TWO-WAY CONCHETE FLOOR SLABS
Ratio of Sides, b/a
FIG. 6. VALUES OF POSITIVE MOMENT CORRECTION FACTORS
l
ILLINOIS ENGINEERING EXPERIMENT STATION
of the distribution constants may be used in order to simplify the
numerical work involved.
Expressions for modified stiffness and carry-over factors for a
number of common cases have been derived by the method described
in Section 16. The cases considered. and the notation used, are
illustrated in Fig. 7. The algebraic expressions are given below for
the constants on a short edge. b. Corresponding expressions applying
to a long edge may be obtained by simply interchanging the sub-
scripts a and b in each case.
Case 0. Basic panel:
Stiffness factor: kb
Carry-over factors: C(' and CG,
Case 1. Opposite edge simply-supported:
k6 = kb(l -Cb2)
Cbý
1 + C&4
Case 2. Adjacent edge simply-supported:
k' = k,(1 - CC,,,)
Cr -- C,,sC,
1 - C'.
1 - C C- -
Case 3. Opposite and adjacent edge simply-supported:
k' "= k'I (1 - " (,)
C"
'" - 4
1+ C",
Case 4. Symmetrical conditions of loading and deformation:
1:, = k,,(1 + C,,)
S ,., = 2
1 - CV,
Case 5. Symmetry plu. adjacent edge simrpliy-xupporled:
',' = k', (1 - (C',,
k / .I+ C "',,
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
The moments on the fixed edges of panels with one or more edges
simply supported may be obtained from the values of fixed-edge
moments by means of the distribution procedure. This calculation
is easily performed as the first step in the solution. (See the illustrative
example in the following section.)
Case /
Case 2
(Case .
Le__Z'-F-ixed Ed ge
----_- Simplj- Supforfed
£a'ie
Ede
Case 3
FIG. 7. NOTATION FOR MODIFIED STIFFNESS AND
CARRY-OVER FACTORS
8. Illustrative Example
To illustrate the application of the distribution procedure, it will
be used to compute moments in the continuous slab shown in Fig. 8.
The loading considered will be w = 100 lb per sq ft, uniformly dis-
tributed over the entire area of the slab. The thickness of the slab
is assumed to be the same for all panels.
26 ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 8. PLAN OF SL.AB FOR ILLUSTRATIVE EXAMPLE
FIG. 9. DISTRIBUTION CON.TANTS FOR ILLr-.TRATIVE EXAMPL.E
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
Since both the structure to be analyzed and the loading are
symmetrical about both centerlines, only six of the fifteen panels need
be considered. Furthermore, in panels (3), (4), (5), and (6), modified
stiffnesses and carry-over factors may be used to take account of
the symmetrical conditions.
The beams supporting the edges of the panels are assumed to be
nondeflecting and to offer no restraint against rotation of the slab;
that is, they have no torsional rigidity. The exterior edges of the
slab are therefore simply supported, and modified distribution con-
stants may be used in the edge panels.
TABLE 2
('CALLATIONS FOR BASIC STIFFNESS FACTORS AND FIXED-EDGE MOMENTS
FOR ILLU'STRATIVE EXAMPLE
SSpan, F.E.M. F.E.M.t
Panel - Edge k , K* -- .f, May,
ft rb' lb
1, 2 0.75 Long 6.25 3 15 0.417 0.0415 -934
Short 7.75 15 0.517 0.0313 -704
3 0.50 Long 5.00 10 0.500 0.0556 -556
Short 7.50 10 0.750 0.0314 -314
4, 5 0.67 Long 5. 78 10 0.578 0. 0464 -464
Slhort 7.67 10 0.766 0.0315 -315
6 I 1.00 All 8.00 10 0.800 0.0290 -290
*N=1.
-t = 100 lb per sq ft.
The particular distribution constants required in the solution of
this problem are indicated by the appropriate symbols on the sketch
in Fig. 9. Basic values of the carry-over factors were determined
from the curves and formulas in Fig. 5, and values of the modified
factors were computed by means of the equations in Section 7. The
resulting numerical values are given in Fig. 10a.
The calculation of the basic stiffness factors is shown in Table 2.
The values of k were obtained from Fig. 3, and the quantity K
was computed for a value of N = 1, since the slab thickness is equal
for all panels. The numerical values of K thus obtained are also
shown in the boxes in Fig. 10a. The distribution factors which deter-
mine the proportion of the unbalanced moment to be distributed
to each panel are written next to the boxes. These factors are com-
puted as in the Cross method of moment distribution for continuous
ILLINOIS ENGINEERING EXPERIMENT STATION
-3/5 -31S5 -3/5 -290 (-209)
+2Z4 + 9-44 + 99 - 5 -24
(-785) - 63 + 6Z (-87 -63 + 65
-315 -7 + 5 + 3 -5 + T3 - /9
+3/1s - / + / - 2 - // + -
0o 0 + 2 +2 + 21- 3 +
- 6 + / - / - - + 3
+ 13 + 13 87 3
-59 - 12 - 39
-259 ('-/4) (-.0O6)
-63 -/3E + 90
D -464 _ -464 F -290
-7-04 -704 -3/4
-299 + 41 +197
- 49 - 8 - 83
- 5/ +149 + 20
-934 +3/7 - 8 - 28
+934 0 -934 -934 + 5 -934 -556
.+ 7 -2/5 -/55 - 55 -4? -47
-/90 - I +' -20/ 0
+ / +122 --01 0
- I + z - /
- 9g -59 - 33 + I -
-779 - 33 -,99 + 25 - ZO -211
(-785) 0 0 (-587) - / + 4 (-209)
- z - 6 - f
(b)-D-OSTR,/BUT/rN - / + 3 - 3
O,- mo1o mS - + 3 0 + /
0 - + 0
(Exact moments - / 0 . -80
i/ p/arenheses.) -1314 (-850)
(-1370)
0 -0 0
+7-04 +-704 +3/4
1,4 -704 -704 c -3/4
FIG. 10. ILLUSTRATIVE COMPUTATION SHEET
I
I
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
beams. For example: on edge BE the total stiffness is 0.369+0.390 =
0.759, of which 0.369 0.759=0.49 is contributed by panel (1); and
0.390 0.759=0.51 by panel (2).
The calculation of the average fixed-edge moment is also illus-
trated in Table 2. Values of the coefficients of M,,,. "wb" were obtained
from Table 1 and Fig. 1.
All necessary distribution constants are given on the sketch in
Fig. 10a, and the complete calculations for edge moments are given
in Fig. 101). The operations carried out in these calculations may be
described step by step as follows:
(a) Write in the average fixed-edge moment for each edge at a
location adjacent to that edge as shown.
(b) Release the restraints on all exterior edges, which are simply
supported in the actual structure, by balancing that moment on
each edge to zero.
(c) Carry over the proper proportion of the balancing moment
from step (b) to the interior edges of each panel. In the corner panel
(1), the carry-over factors are modified so that moments need not
be carried over to the adjacent simply-supported edge.
From this point forward, the use of modified distribution factors
for the interior edges obviates the carrying over of any moments
to the simply-supported edges.
(d) Release the restraints on edge BE and distribute the un-
balanced moment of -250 to the two panels in proportion to the
distribution factors, thus: + 122, or 49 percent, to panel (1); -128,
or 51 percent, to panel (2).
(e) Carry over the distributed moments to the remaining interior
edges of panels (1) and (2). For example, in panel (2) carry over
-128 X -0.32 = +41 to edge EF, and -128 X -0.23 = +29 to
edge CF.
(f) Repeat the processes of steps (d) and (e) for edges CF, DE,
EF, FF', EE, and FF in turn, first balancing and then carrying
over at each edge before proceeding to the next edge.
(g) Start again at edge BE, and repeat the process at successive
edges until no unbalanced moment exists on any edge.
(h) Add all the moments in each panel at each edge. These are
the desired average moment on the edges of the continuous slab.
In this problem the procedure has been followed of distributing
moments at each edge successively. In other problems it may be
more convenient to release all edges simultaneously. In either case
the procedure has a direct parallel with that used for continuous
beams or frames.
30 ILLINOIS ENGINEERING EXPERIMENT STATION
The calculation of average positive moment on sections in the
interior of each panel is illustrated in Fig. 11. The positive moment
correction factors, F, obtained from Fig. 6, are given on the small
sketches for each group of panels having the same values of b/a.
Average positive moments, computed from the coefficients in Table 1
and Fig. 2, are written on a sketch of each panel adjacent to the
section to which they refer. Corrections due to the edge moments,
obtained as the product of those moments and the appropriate
factor, F, are written separately for the moments on the long and
short edges. The net average positive moment in each direction is
obtained as the sum of the positive moment for the simply-supported
slab and the corrections due to the edge conditions.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
0/9 -
+1 I 1 ++
'P+ri +
0/8-
. Fi ffif £?/ff -9>
N L--- |L| -rL-
IX ____ S____4_ v
+1 1 +\+
-NM< ^»
O'J '-N
~l'-
'\l ~-~N
+ I 4+
N
26'2-
~C)
~(l~I~ I
~ I ~
N I
+1
+1 I +,t
%2 +> +N,
-I-II 1-
+1 I +4
'~ ~'J ~
+
'I-
to
~a ~
+4
sw
I
I/-
!3'Q|--|--1 |--r-^v
^ ^ ^
^'l' _^_ _^;;
S2 to l/
§ | ^
WW L-^-^
11I.1LI1NO ENt;INEElRING EXPERIMENT STATION
III. DEVELOPMENT OF THE DISTRIBUTION PROCEDURE
9. Introduction
The distribution procedure is not exact, chiefly because of the
approximate nature of the stiffnesses, carry-over factors, and posi-
tive moment correction factors, as applied to average moments. The
other constants in the procedure, fixed-edge moments and moments
in the simply-supported slab, are obtained from exact solutions and
do not contribute to the approximate nature of the procedure, except
insofar as average moments are used. The method of distributing
moments is not in itself approximate, but is of course no more
accurate than the constants involved in its use.
It is tile purpose of this chapter to describe in detail the manner
in which the final values of the various distribution constants were
obtained, and to establish their validity and degree of approximation
by applying the distribution procedure to the solution of certain
problems for which the exact answers are known.
In the derivation of sets of elastic constants to be used in an
approximate procedure for analyzing continuous plates, it is neces-
sary first to make some decision regarding the manner in which
moments and rotations are distributed along the edges of a panel.
One problem in tile development of the procedure, therefore, was to
determine the extent to which the various constants were affected
by changes in the assumed shapes of the distribution curves. This
was done by first investigating two approximate procedures that had
previously been developed for the analysis of continuous plates.'
Each of these procedures is based on a different set of assumptions
regarding the shapes of the curves of moment and rotation, and
from the results of each set of assumptions it was possible to derive
elastic constants corresponding to the stiffness and carry-over factors
used in the moment distribution procedure.
The next step was the derivation of another set of stiffness and
carry-over factors by means of a semi-rational procedure based on
the moments in uniformly loaded isolated rectangular plates. These
factors corresponded to still another set of assumptions regarding the
manner in which the moments and slopes are distributed along an
edge. The three sets of constants thus obtained were compared and
the effects of variations in the assumptions were ascertained.
' After the completion of the thesis on which this bulletin is based, a third proceduie
came to the authors' attention. This method is similar to that described in Chapter II in that
the moment distribution concept is used, but the underlying assumptions are different. In spite
of these differences, the final results in the form of stiffnesses and carry-over factors are not
greatly different froi those proposed herein. The reference to this paper is "En Metode for
Tilnaermet Beregning av Kontinu-rlioe Toveisplater." Ib Kndl Engelhreth, Betoing, 1945, Vol. 30,
No. 2. pp. 99-115.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
It was then necessary to determine the effect of variations in the
constuants on the moments computed by means of the distribution
procedure. A study of this problem revealed that the moments were
only slightly affected by relatively large differences in the constants
obtained for the different assumptions. Consequently, a set of con-
stants for use in the distribution procedure was selected, and was
then checked by using them in the analysis of continuous slabs for
which the correct moments were known.
The derivation and checking of the positive moment correction
factors was an entirely separate step, described in Section 17.
FIG. 12. NOTATION tSED IN DERIVATION OF
BITTNER AND MA.UIIn-PAN C'ONTANTS.
10. Bittner's Method of Analysis
This procedure was proposed by E. Bittner' in 1938 for the
calculation of moments in continuous slabs. The basis of the method
is the expression of the rotations on each edge of a panel in terms of
the moments on all edges and the applied load. For continuous slabs
the resultant slope in one panel must be equal to the resultant slope
on the same edge in an adjacent panel. These conditions, plus the
boundary conditions at the edges of the slab, permit the solution of
the various equations for the unknown moments. This procedure is
essentially algebraic, in contrast to the numerical procedure de-
scribed in Chapter II of this bulletin.
The only part of Bittner's procedure that is of direct interest
here relates to the elastic constants used to express the edge rotations
in terms of edge moments. Both the assumptions made in the deriva-
tion of these factors and their numerical values are of interest.
Bittner's approach to this problem is as follows: Consider a rec-
tangular panel supported on all edges as shown in Fig. 12. If a single
' E. Bittner. "''Monententafeln und Einflussfliichen fur krenzweise bewehrte Eisenbelon-
Ilatten." Julius Springer. Vienna. 1938.
ILLINOIS ENGINEERING EXPERIMENT STATION
sine wave of moment, M1 sin -, is applied on edge (1), the rota-
b ry
tion of edge (1) is also a single sine wave, '1 sin -, in which
b
41 = O'M1
and Oy is the designation used by Bittner to represent the relation
between F1i and M, on the short edge of a panel. The rotation on edge
(2) is ('2 sin --, wherein
b
o2 = 3vM,.
The rotations on edges (3) and (4) are not single sine waves and
therefore cannot be expressed as simple functions of MI except in
an approximate manner. This Bittner does by first expressing the
actual rotations on edges (3) and (4) as a sine series, then considering
only the first term of that series. The resulting rotations are thus
rx TX
P3 sin - and (4 sin -, wherein
a a
4)3 = 04 = OxM1.
The corresponding expressions for moment applied on a long edge
such as (3) or (4) may be obtained from the above expressions by
interchanging the subscripts x and y, and a and b.
General expressions for single sine waves of rotation on each edge
due to single sine waves of moment on all four edges may be written
as follows:
e1 = #,M2 + 'M1 + Oy.(M3 + M4)
'N = I.Ma3 + 'M4 + 3x,(Ml + Ms)
D4 = #ýM4 + ff'M3 + ly(M1 + M2).
The constants Oy and #, may be considered as inverted stiffness
factors relating single sine waves of moment and rotation on the
same edge of a simply-supported panel. Thus
M3 1 27XM
4'3 . a
wherein X is the stiffness factor for which numerical values are given
by Bittner in his Table 1 for values of a/b (1/h in his notation)
varying from 0 to 10.0. It may be noted that Bittner's stiffness
factor, X, is directly related to the modified stiffness, S, given by
Newmark in Bulletin 304.' The relation between numerical values
a N. M. Newmark, 'A Distribution Procedure for the Analysis of Slabs Continuous over
Flexible Beams," Univ. of Ill. Eng. Exp. Sta. Bul. 304. 1938.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS 35
of these constants is as follows:
8
2 -X = C b-,
where C, is given in Table 1 of Bulletin 304, and s and b are the
lengths of the sides of the panel under consideration.
Constants which may be considered as carry-over factors for rota-
tions in a simply-supported slab may be derived from ratios of the 3-
constants in Eq. (1). This is done below, using Bittner's notation.
- -v xy - =x pyx.
The various p-constants above are given by Bittner in Table 2, for
values of a b (1,,/l in his notation) ranging from 1.0 to 1.5. Again,
it may be noted that the carry-overs to an opposite edge, p, and
,p,, are identical with the carry-over factor, k, given by Newmark in
Table 2 of Bulletin 304.
The stiffnesses and carry-over factors derived by Bittner apply
to a simply-supported panel subjected to a single sine wave of mo-
ment on one edge and subject to certain assumptions regarding the
distribution of rotation on the adjacent edges. It is possible, however,
to derive from these constants another set, referring to a fixed-
edge slab with a unit rotation applied to one edge. Such constants
would be equivalent to those used in the distribution procedure
developed herein.
The equivalent stiffness and carry-over factors may be obtained
from Bittner's constants in the following manner: First, consider
only the constants corresponding to a moment applied on the short
edge of a panel. If in Eq. (1)
(I+ = 1
then 42 = 43 = 44 = 0,
M,
M, Cb
-= Cbb
MM
M, M
= M, = Kb= kb-,
1Pi b
ILLINOIS ENGINEERING EXPERIMENT STATION
and the following expressions are obtained:
21 -- -
C ba-
1 + -
Cbb = - 1- - 1 "- *Cb.
2rX
kb =-------
1 + ~pyCbb + 2ioPxCbo
The constants Cab, Can, and k., referring to a long edge, may be
obtained by a similar calculation.
TABLE 3
EQUIVALENT STIFFNESS AND CARRY-OVER FACTORS COMPUTED
FROM BITTNER CONSTANTS
b/a
Constant
0.50 0.67 0.75 1.00
Stiffness Factors: k. 4.92 5.72 6.21 7.79
ka 6.84 7.18 7.39 7.79
Carry-over Factors: C.. -0.317 -0.199 -0.140 -0.017
C.a -0.274 -0.304 -0.307 -0.268
Cbb +0.062 +0.062 +0.053 -0.017
CM, -0.099 -0.161 -0.193 -0.268
Numerical values of the equivalent factors computed from Bitt-
ner's constants are given in Table 3. It will be noted that values are
given for b/a = 0.5, although Bittner does not tabulate the p-values
for b/a less than 0.67. For this case px and ,y were obtained from
their equality with the constants in Bulletin 304, and values of
(P., and py, were obtained approximately by extrapolation from
plotted data.
The values given in Table 3 are simply one of a number of
possible sets of constants describing the behavior of a fixed-edge
slab with a unit rotation applied on one edge. The values of these
constants depend to some extent on the manner in which the mo-
ments and rotations are distributed along the various edges of the
panels. In Bittner's solutions, both of these quantities are assumed
to be distributed as a single sine wave on each edge.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
11. Method of Maugh and Pan
This approximate method of analysis for continuous slabs was
presented by L. C. Maugh and C. W. Pan' in 1941. It is similar in
principle to Bittner's procedure and involves algebraic expressions
for edge rotations in terms of edge moments. Somewhat different
assumptions were made, however, in treating the rotations on the
adjacent edges, and the carry-over factors to those edges are thus
different from those obtained by Bittner. The carry-over factors to
the opposite edge and the stiffness factors are the same as those
given in the preceding section.
The chief assumptions made by Maugh and Pan are 1) that all
edge moments are distributed as single sine waves and 2) that con-
tinuity between various panels will be maintained only at the middle
of each edge. The first assumption is also made by Bittner, but the
second is different, as may be seen from the following discussion.
Consider the panel in Fig. 12 with a single sine wave of moment
ry
M1 sin-- acting on edge (1). The rotation on edge (1) is also a
b 7ry
single sine wave, (1 sin --, and the rotation at the middle of the
b
edge, 41, is given by the expression
b
)i = ab -- M1,
N
wherein ab is the authors' designation for the stiffness factor involved.
Similarly, the rotation at the middle of edge (2) may be expressed as
b
2 = b l M1,
N
in which 0b is Maugh and Pan's notation and should not be confused
with the #-terms used by Bittner. In the above expressions both
the moments and the rotations are distributed as single sine waves,
and the results obtained are thus the same as those obtained by
Bittner. The relations are as follows:
b
(Maugh) ab- = (Bittner) /,,
N
b
(Maugh) fb-- = (Bittner) fly.
1 I.. M a n C. C ' V. an )d ' . I'!lrl . 1Mjl ltlt ill C(llltinltn llm ,s iRv ,allillra la s ll n Rigid Sllp-
i.,rts." Tra-.. A4 E'K. V\,l. 107 (1942), p. 1118.
ILLINOIS ENGINEERING EXPERIMENT STATION
The rotations on edges (3) and (4) are not distributed as a single
sine wave. In this case, Bittner expanded the rotation into a sine
series and dealt only with the first term. Maugh and Pan, however,
retain the actual distribution but consider only the value of the
rotation at the middle of the edge. These rotations are expressed
by the following equation:
b
t3 = 4 Yb - M1.
N
Maugh and Pan's constant 7Yb is thus different from Bittner's con-
stant 0,.
All the above expressions apply only for a moment on a short
edge. Similar expressions for moment on a long edge may be obtained
by interchanging a and b.
The general expressions corresponding to Eq. (1) of Section 11
are as follows:
b b a
'1 = ab -- M + b-- M2 + - (M3 + M4)
N N N
b b a
'2 = a M+ b + .- (M3 + M4)
N N N
a a b (2)
I3 = a--M3a -- M4 +b --(M +M2)
N N N
a a b
4 = a-- M4 + a M3 +b - (M +M2).
N N N
Expressions for equivalent stiffness and carry-over factors were
obtained from the above equations in a manner identical with that
described in the preceding section in connection with Eq. (1). The
expressions for constants referring to a short edge are:
b ^Yb(ab - b)
Cba = ------------
a2 7a7b - aib(a + |a)
a (aa + )
Cbb = -1 - --Cba
b Yb
1
kb =
a
ab + 6bCbb + 27Ya - Cb
Constants for a long edge may be obtained in a similar manner.
Numerical values of the equivalent stiffness and carry-over factors
computed from Maugh and Pan's constants are given in Table 4.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
These values represent the results of a set of assumptions different
from those made by Bittner, since in their solution for the simply-
supported slab Maugh and Pan have assumed a single sine wave
distribution of moment and rotation on only two edges of a panel
and have considered an unsymmetrical distribution on the other two
edges. The corresponding distributions of both moment and rotation
for the fixed slab are not single-sine waves as was the case for Bitt-
ner's solution. Although only one pair of constants for the simply-
supported slab was affected by the difference in the assumptions, all
values in Tables 3 and 4 for the fixed-edge slab are changed, since
all of the constants for the simply-supported slab enter into each
expression for the fixed-edge slab.
TABLE 4
EQUIVALENT STIFFNESS AND CARRY-OVER FACTORS COMPUTED FROM
MAUGH AND PAN CONSTANTS
b/a
Constant
0.50 0.67 0.75 1.00
Stiffness Factors: k. 4.80 5.59 6.03 7.46
kh 6.60 6.97 7.13 7.46
Carry-over Factors: C. -0.346 -0.231 -0.178 -0.062
C.6 -0.247 -0.278 -0.275 -0.227
Cnb +0.031 +0.032 +0.020 -0.062
Cs. -0.063 -0.127 -0.159 -0.227
12. Derivation of Stiffness Factors
The stiffness of a beam is defined as the moment corresponding
to a unit rotation at one end of the beam, the other end being fixed.
For the purposes of this bulletin the stiffness of a slab is defined in
a similar manner, as the moment corresponding to a unit rotation
on one edge of the slab, all other edges being fixed. The determina-
tion of this stiffness, however, is not as simple for a slab as for a
beam, since a question immediately arises regarding the distribution
of rotation and moment on the edge being considered. The procedure
followed by Bittner is equivalent to assuming a single sine wave
distribution for both moment and rotation; the method used by
Maugh and Pan results in an unknown distribution of rotation when
applied to a fixed-edge panel.
The method finally adopted herein for determining the stiffness
factors for a slab is based on the moments and rotations of uniformly
loaded plates with fixed edges. It may be described as follows:
Consider the plate shown in Fig. 12, with edges (2), (3), and (4)
ILLINOIS ENGINEERING EXPERIMENT STATION
fixed and edge (1) simply-supported. The rotation of edge (1) due
to a uniform load w on the plate may then be computed. (Such
values are given in Section 30, Appendix A.) If a rotation equal and
opposite to that for the simply-supported edge is applied to edge (1),
the corresponding moment is equal to the moment on edge (1) of a
uniformly loaded fixed-edge plate. This moment is given in Table 1
and in Section 31, Appendix A.
In effect, the above procedure consists of applying a known rota-
tion to the simply-supported edge of a plate fixed on three edges,
and determining the moment produced on that edge as a result of
the applied rotation. In accordance with the assumptions of the
distribution procedure, only average rotations and moments are
considered. Specifically, the rotation applied is the average rotation
on the simply-supported edge of a uniformly loaded plate fixed on
three edges, and the moment produced is the average moment on
the corresponding edge of a uniformly loaded plate fixed on all edges.
For example, consider a plate having b/a = 0.5, and fixed on edges (2),
(3), and (4) of Fig. 12. The average rotation on the simply-supported
short edge (1) due to a uniform load w is
a b
'av = 0.0019 - wb2 = 0.0038 -- wb2.
N N
The average moment on the short edge of a uniformly loaded plate
fixed on all edges is
May = 0.0314wb-'.
The stiffness on the short edge is thus
May N
Kb =-- = 8.25--
iavy b
and the stiffness factor kb = 8.25.
Data are available in Appendix A to make similar calculations
for the long edge of a plate having b/a =0.5 and for a plate having
b/a = 1.0. The calculations for the stiffness factors k are shown in
Table 5. To distinguish this set of constants from the simplified
constants discussed subsequently, these values will be referred to as
the "derived" stiffness factors.
The stiffness factors from Table 5 are plotted versus b/a in
Fig. 13, together with the corresponding values based on Bittner's
constants (Table 3) and Maugh's constants (Table 4). The stiffness
factors obtained by the various procedures are not greatly different,
the maximum difference being about 20 percent, for kb at b/a =0.5.
There is even less difference in the relative stiffnesses for adjacent
panels as determined from the three sets of constants.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
CALCULATION OF
TABLE 5
DERIVED STIFFNESS FACTORS
b/a Stiffness -_ M. -P- k
b Factor ubw wb2
1.0 kb 0.0290 0.0035 - 8.30
0.5 kb 0.0314 0.0038 - 8.25
0.5 k. 0.0556 0.0110-- 5.06
N
Ratio of Sides, b/a'
FIG. 13. COMPARISON OF STIFFNESS FACTORS
13. Derivation of Carry-Over Factors
The manner of obtaining the "derived" carry-over factors is
described in this section. For the purpose of this bulletin the carry-
over factors for average moments are defined in a manner analogous
to that used for a beam. Consider a rectangular panel, fixed on three
edges and simply-supported on the fourth. If a moment is applied
to the simply-supported edge, corresponding moments are induced
at each of the fixed edges. The ratios of these induced moments to
the applied moment are the carry-over factors.
ILLINOIS ENGINEERING EXPERIMENT STATION
The problem of what distribution of moment should be considered
on the various edges is treated here, as iii the calculation of stiffness
factors, by considering the moments in uniformly loaded plates. The
panel with three edges fixed is assumed to he uniformly loaded, and
the moments on the fixed edges are determined from Section 30,
Appendix A. A moment of just sufficient magnitude to produce the
condition of all edges fixed is then applied to the simply-supported
TABLE 6A
CALCULATION OF DERIVED CARRY-OVER FACTORS FOR IMOMENT
ON A LONG EDGE
SM, - M.,' C 1. - M - Jf'
C< Mo C =M. * f-
All values for moments are M,,/wb2
- Mf.' - Mbs
0.0726 0.0315
(0.0654) 0.0315
(0.0557) 0.0315
0.0482 (0.0314)
(0.0440) 0.0311
0.0307 0.0290
- .!,'
0.0487
(0.0484)
(0.0475)
0. 0455
(0.0440)
0.0381
C.O Cab
-0.306 -0.309
-0.262 -0.326
-0.200 -0.345
-0.156 -0.338
-0.131 -0.332
-0.059 -0.314
Values in parentheses obtained by interpolation from data in Appendix A.
TABLE 6B
CALCULATION OF DERIVED CARRY-OVER FACTORS FOR MOMENT
ON A SHORT EDGE
Mi -b i" M. -- M"
All values for moments are M1v./wb2
b/a -. b - Mb" - M, - M." Cbb Cbo
0.500 0.0315 0.0314 0.0556 0.0610 +0.003 -0.172
0.571 0.0315 (0.0314) 0.0518 (0.0580) +0.003 -0.197
0.667 0.0315 (0.0314) 0.0464 (0.0532) +0.003 -0.224
0.750 (0.0314) 0.0314 (0.0417) 0.0495 0 -0.248
0.800 0.0311 (0.0312) 0.0389 (0.0470) -0.003 -0.262
1.000 0.0290 0.0307 0.0290 0.0381 -0.059 -0.314
Values in parentheses obtained by interpolation from data in Appendix A.
b/a
0.500
0.571
0.667
0.750
0.800
1.000
-M.
0.0556
0.0518
0.0464
(0.0417)
0.0389
0.0290
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
Ra7tio of Sides, b/a"
FIc. 14. COMPARISON OF CARRY-OVER FACTORS
edge. These moments are obtained from Section 31. The carry-over
factors are then determined as the differences in average edge mo-
ments for three edges fixed and for all edges fixed, divided by the
moment applied to the simply-supported edge (actually the mo-
ment for all edges fixed).
The calculation of the derived carry-over factors by the above
procedure is illustrated in Tables 6a and 6b. The notation used is
indicated on the sketches accompanying each table. Expressions for
the various carry-over factors are indicated, and the values of the
necessary moments are given in the tables.
Values of the carry-over factors given in Table 6 are plotted
versus b a in Fig. 14, together with the corresponding factors based
on Bittner's and Maugh's constants as given in Tables 3 and 4,.
respectively. Values of the derived constants for b a less than 0.5
were computed from the following approximate expressions for the
average moments. The notation is that of Table 6.
ILLINOIS ENGINEERING EXPERIMENT STATION
- MA'wb2 -- 0.0833 - 0.0554b a
- Mbwb2 0.0315
- Ma wb2 -0.1250 - 0.1048b a
-M'b wb2 0.0487
-M' wb2 0.0833 - 0.0446b/a.
The agreement between the three sets of carry-over factors in
Fig. 14 is not so good as that for the stiffness factors in Fig. 13. As
might be expected from the nature of the assumptions, the greatest
differences occur for the lateral carry-over factors, Cba and Cab. The
differences in values of Cbb are not particularly significant, since this
constant is small. The actual importance of the differences, however,
can be assessed only by using the constants in the distribution
procedure for the solution of problems to which the correct answers
are known. Some of the results obtained from such solutions are
discussed in the following sections.
14. Simplified Distribution Constants
In the preceding sections of this chapter the manner of obtaining
three different sets of distribution constants, based on three different
sets of assumptions, has been described. It may be seen from Figs.
13 and 14 that there are fairly large variations between the values
thus obtained. The problem at this point is to determine the extent
to which these variations affect the moments computed by means
of the distribution procedure, and to decide which values of the con-
stants shall be used. To this end, the distribution procedure using
each set of constants in turn was applied to the analysis of three
continuous slabs for which the exact moments were known. The
three slabs analyzed were designated I, II, and III; the exact solu-
tions are given in Sections 33, 34, and 35, respectively, of Appendix B.
Plans of the slabs are given in Fig. 16.
The maximum variation between the average moments computed
using the "Bittner," "Maugh," and "derived" constants was about
10 percent, plus or minus, with two exceptions. The exceptions were
the moments on edge EE of slab II and on edge DD of slab III.
In both these cases, use of the derived constants resulted in average
moments considerably smaller than the correct values. This de-
ficiency of the procedure was not considered unduly important,
however, since the moments on the edges in question were small
compared to the other moments in the structure.
MOMENTS IN TWO-WAY CONCRETE FL.OOR SIABS
The results of hlie study mentioned above indicated that rela-
tively large variations in the distribution constants produced only
small variations in the average moments. The three sets of constants
were replaced by a single set of modified constants chosen to represent
roughly the averages of the original three sets:
- C,,, = 0.560 - 0.535 b a, for b a = 0.3-1.0
1
- Cbb = - (b'a - 0.6), for b a = 0.6-1.0
16
= 0, for b/a = 0 -0.6
--CGb = 0.200 + 0.180 b/a, for b/a = 0 -0.7
= 0.433 - 0.153 b/a, for b/a = 0.7-1.0
- Cba = 0.280 b/a, for b/a = 0 -1.0
ký = 4 [1 + (b/a)2]
b
kb = 7 + -.
a
The above modified constants were then used for the solution of
the three problems previously mentioned. The average moments thus
obtained were little different from those computed with the derived
constants or with either of the other sets. In general the modified con-
stants gave results slightly better than did the derived factors. This
first attempt at simplification was therefore successful in that it re-
duced the complexity of the relation between the distribution con-
stants and b/a without affecting the accuracy of the procedure.
It was finally decided to attempt an even greater modification of
the constants in an attempt to secure even more simple relations
with b/a. The equations for the stiffness factors were not changed;
the expressions for the carry-over factors are as follows:
-Coa = 0.60 (1 - b/a), but not more than 0.5
- Cb = 0
-Cab = 0.30
- Chb = 0.30 b/a.
Values of the constants determined from the above equations are
given in Table 1 and Fig. 5. Values of the stiffness factors from the
equations given previously are found in Table 1 and Fig. 3, and are
also indicated by open circles in Fig. 13.
ILLINOIS ENGINEERING EXPERIMENT STATION
A/ll Momens - /V,./wh
-377
(-.99)
0.99
4p,,rox. - - 90
Exa'ct..- (-307)
rf--o-...... 0.6s
-464
(-48/)
0.96
(-572)
0.99
/
/
/
/
/
/
-603 /
-6/0/)
0.99 /
/
/
/
/
/
/
/
/
-6SO 0
(-662)
0.98
/
-05
-3/4
(-3/4)
/.00
~ed
-793
(-796)
4.00
FIG. 15. COMPARISON OF APPROXIMATE AND EXACT MOMENTS
FOR UNIFORMLY LOADED SINGLE PANELS
15. Verification of Distribution Constants
The simplified distribution constants derived in the preceding sec-
tion were checked in two ways: 1) the carry-over factors alone were
used to compute the average edge moments in uniformly loaded single
-723
(-726)
0.99
-48/
(-487)
0.99
/
/
/
/
/
/
-92s /
/
(-.906);
1.02 /
/
/
/
/
/
-8/9
C-8/3s)
1.0/
Sulppormef
'Edg'e
-906
(-5/2)
/. /2
//lll///.
J//J/1////
I
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
A 8
Slab Z
A B B
Slab 27 Slab r7
Fl(;. 16. PLANS OF CONTINUOUS SLABS I'SED IN CHECK
OF DISTRIBUTION CONSTANTS
panels with one, two, or three edges fixed; 2) the entire distribution
procedure, utilizing the simplified constants, was applied to the
analysis of uniformly loaded continuous slabs for which the exact
moments were known. These two procedures are described below.
The first check was made by using the simplified carry-over
factors to determine the average edge moments in uniformly loaded
single panels with one, two, or three edges fixed. Panels having b/a
ratios of 1.0 and 0.5 were used. The procedure was as follows:
Average edge moments for a panel with all four edges fixed were
obtained from Table 1. The various edges were then successively
released and the appropriate moment was carried over to the other
edges by means of the simplified carry-over factors of Section 14
and Table 1. The results are given in Fig. 15, in which the various
panels and edge conditions considered are shown by sketches. Cross-
hatching is used to indicate a fixed edge and all other edges are
simply-supported. The approximate moments determined from the
distribution procedure are given for each fixed edge, together with
the exact average moments in parentheses and the ratio of approxi-
mate to exact moment. The exact moments were obtained from the
data in Appendix A.
The agreement between approximate and exact moments in
Fig. 15 is quite good, generally within 5 percent plus or minus. An
exception occurs for the rectangular panel having one short edge
fixed, for which the distribution procedure gives a moment 12
percent too high.
The second check involved the entire distribution procedure and
consisted of the analysis of three continuous slabs, plans of which
are given in Fig. 16. All slabs were considered to carry a uniform
load, W, of 100 lb per sq ft over their entire area. The exact solu-
tions for moments in these slabs are described in detail in Appendix B.
ILLINOIS ENGINEERING EXPERIMENT STATION
Slab II was used for the illustrative example in Section 8, and its solu-
tion by means of the distribution procedure is given in Figs. 9 and 10.
Comparisons of the approximate and exact average edge mo-
ments for the three slabs are given in Table 7. The agreement is
quite good except at the two locations mentioned previously, edge
EE of Slab II and edge DD of Slab III, for which the ratios of ap-
proximate to exact average moments are 0.77 and 0.71 respectively.
It should be noted, however, that the moments on these edges are
small compared to the other moments in the structures. A study of
the behavior of the slabs suggests that discrepancies of this magnitude
are the result of conditions which also result in small moments on
the edges in question.
In general, the agreement between approximate and exact mo-
ments indicated in Fig. 15 and Table 7 was believed to be sufficiently
good to warrant the use of the simplified constants and, in addition,
to constitute adequate verification of the distribution procedure as a
means of computing average moments in continuous slabs.
16. Calculation of Modified Constants
Expressions for modified distribution constants for use in cases
of symmetry or in panels with simply-supported edges have been
given in Section 7. The manner of obtaining these expressions is
described in this section. Since a single example should be sufficient
to explain the procedure, the constants K,', C,, and C" will be de-
rived for the short edge of a panel in which an adjacent long edge is
simply-supported. This is Case 2 of Section 7.
Consider the rectangular panel of Fig. 12. The average moment
on each edge may be expressed in terms of the average rotations on
all edges and the distribution constants K and C as follows:
M, = Kb, + CbbKb2 + CbK. (43 + D4)
M2 = Kb02 + CbbKbDl + CabKa (?3 + 44)
(3)
M3 = KA3 + CooKA4 + CbaKb ('l + 42)
M4 = KJ4 + CaKA3 + CbaKb (41 + 42).
To determine the modified constants for the case being considered
apply a unit rotation to edge (1), edges (2) and (3) being considered
fixed, and edge (4) simply-supported. These conditions are stated
as follows: 1
i 1 = 1
42 =P3 = 0
M4 = 0.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
The values given above are then substituted into Equations (3) and
-11, M2, and 313 are solved for. The modified constants desired are
then determined as follows:
I,1
C'11 =
TABLE 7
COMPARISON OF APPROXIMATE AND EXACT EDGE MOMENTS FOR UNIFORMLY
LOADED CONTINUOUS SLABS
See Fig. 16 for notation. All values are average moments in pounds for a uniform load w, over
entire slab of 100 lb per sq ft.
* Using distribution procedure with simplified constants (Table 1).
t For method of calculation see Appendix B.
The resulting algebraic expressions for these constants are given
in Section 7.
A similar procedure was followed in calculating all the modified
constants for which expressions are given in Section 7.
17. Derivation of Positive Moment Factors
The positive moment correction factors have been discussed
briefly in Section 6, and the numerical values finally adopted for use
with the distribution procedure are given in Table 1 and Fig. 6.
These factors are used to correct the average positive moments in
the interior of a simply-supported panel for the effect of edge mo-
ments resulting from continuity.
- M,
ILLINOIS ENGINEERING EXPERIMENT STATION
It was not considered feasible to derive the positive moment
factors by direct calculation, because of the wide range of edge
conditions for which they must be applicable and because of the
manner in which the location of the maximum positive moment in
a given span depends on the edge conditions. Consequently a cut-
and-try procedure was adopted, based first on the exact moments
given in Appendix A for single panels with various edge conditions,
and finally on the exact solutions for continuous slabs given in
Appendix B. Steps followed in the derivation of these factors are
outlined below.
The first step involved the consideration of single rectangular
panels having values of b/a equal to 0.5 to 1.0 and having various
numbers of edges fixed. Exact values of the average edge moments
and the average positive moments in the interior of such panels are
given in Appendix A. A panel simply-supported on all edges was
considered first. Various edges were then fixed, and the relations
between the resulting edge moments and the changes in the average
positive moments were noted. These relations were used to obtain
preliminary values of the desired factors for b/a equal to 0.5 and 1.0.
The second step consisted of using the tentative factors obtained
from the single panels to calculate the positive moments in panels
of the continuous slabs having appropriate values of b/a. In all cases
the value of average positive moment used was that for a section
having the maximum moment. Except for nearly symmetrical condi-
tions, the maximum positive moment does not occur at midspan.
As a result of these checks with exact moments for continuous slabs,
the tentative values were revised and modified. In making these
modifications, greater weight was given to the results for the con-
tinuous slabs than to those for the single panels.
The two steps described above furnished values of the positive
moment factors for b/a equal to 0.5 and 1.0. The third step involved
the determination of these factors for intermediate values of b/a.
This was done first by rough interpolation from curves plotted for
a ranfge of b/a from 0.5 to 2.0. Tentative values were then taken
from such curves for b/a equal to 0.67 and 0.75 and were applied
to the calculation of positive moments in the panels of Slab II having
the corresponding values of b/a. Revisions in the tentative values
having been made as a result of these calculations, the revised values
were used to correct the curves referred to above. The adjusted
curves are shown by solid lines on Fig. 17.
Calculations made with several different sets of positive moment
factors, some of which differed by large amounts, indicated that
MOMENTS IN TWO-WAY CONCRETE FLOOR sIABS
small variations in these factors had no serious effect on the positive
moments obtained. With this fact in mind, it was decided to attempt
a simplification of the relations between the positive moment factors
and b a. The simplified values are shown by dash lines on Fig. 17.
These values are the same as those given in Table 1 and Fig. 6. As
a final step the simplified factors were used to compute moments
in all panels of the three continuous slabs. The results are given
in Table 8.
K
K
N.
Ratio of Siades, h/a
FI(. 17. PRELIMINARY VALUES OF POSITIVE IMOMENT
CORRECTION FACTORS
The approximate moments in Table 8 were obtained entirely by
means of the distribution procedure, using the constants given in
Table 1. The calculations for positive moments in the panels of
Slab II are illustrated in Fig. 11, and described in Section 8 in con-
nection with the illustrative example. The exact average moments
were obtained from the solutions in Appendix B. It may be noted
from the data in Table 8 that the approximate procedure almost
always results in positive moments that are larger than the correct
ILLINOIS ENGINEERING EXPERIMENT STATION
values. This is the result of a deliberate attempt to be conservative
in this phase of the procedure, partly because of the greater impor-
tance of the positive moments and partly to allow for some uncer-
tainty regarding the location of the section for maximum moment
in slabs with different edge conditions. In general the moments ob-
tained with the simplified constants are slightly greater than those
obtained with the constants indicated by the solid curves in Fig. 17.
TABLE 8
COMPARISON OF APPROXIMATE AND EXACT POSITIVE MOMENTS FOR
UNIFORMLY LOADED CONTINUOUS SLABS
See Fig. 16 for plans of slabs and notation. All values are average moments in pounds for a uni-
form load w of 100 lb per sq ft over entire slab. Edge moments obtained by distribution procedure
using approximate constants (see Table 7).
b Approximate Exact Ratio
Slab Panel - Span Average Average (5)
a Moment Moment* (6)
(1) (2) (3) (4) (5) (6) (7)
I DDDD 1.0 ..... +122 +104 1.17
BBDD 1.0 BB +153 +143 1.07
DD +103 + 89 1.16
ABCD 1.0 ..... +156 +153 1.02
II ABDE 0.75 Long +293 +282 1.04
Short +586 +636 0.92
BCEF 0.75 Long +219 +185 1.18
Short +491 +487 1.01
CC'FF' 0.50 Long + 78 + 64 1.22
Short +214 +207 1.03
DEDE 0.67 Long + 52 + 41 1.27
Short +168 +148 1.14
EFEF 0.67 Long + 73 + 64 1.14
Short +191 +160 1.19
FF'FF' 1.0 FF' +144 +133 1.08
FF +129 +110 1.17
III ABCD 1.0 ..... +708 +707 1.00
BBDD 0.5 Long + 44 + 25 ....
Short + 45 - 35
DDDD 1.0 ..... +196 +164 1.20
* Based on data in Appendix B.
Moreover, it should be noted that
the excess of approximate over
exact moment is due in part to the fact that the edge moments ob-
tained by the distribution procedure are usually a little on the low
side, as may be verified by reference to Table 7.
On the basis of the data in Table 8 it was concluded that the sim-
plified positive moment correction factors could be used with the
distribution procedure to give conservatively high values of the
positive moments. This conclusion was also borne out by calculations
of positive moments in single panels with various edge conditions,
the results of which are not given here.
MOMEINTS IN TWO-WAY CONCRETE FLOOR SLABS
IV. APPLICATION OF DISTRIBUTION PROCEDURE TO
CALCULATION OF IMOMENTS IN TWO-1WAY
REINFORCED CONCRETE SLABS
18. Introduction
Although the distribution procedure described in the preceding
chapters was developed primarily for the calculation of moments
in two-way slabs of reinforced concrete, it is recognized that the
usual assumptions of elasticity, isotropy, and homogeneity are not
satisfied completely by a reinforced concrete slab at working stresses.
There is some precedent, however, for the use of a theory based on
these assumptions for the analysis of reinforced concrete structures.1
Although the elastic theory is used primarily because nothing better
is available, comparisons with test results have indicated that the
computed moments and stresses are almost without exception on
the safe side.
In applying the distribution procedure to the calculation of
moments in reinforced concrete slabs supported on reinforced concrete
beams cast monolithically with the slabs, two factors are introduced
which have an important effect on the moments. The first of these
is the restraint to rotation of the edges of the slab resulting from the
torsional rigidity of the supporting beams. This effect, quite important
for certain types of loading, is considered in some detail in this
chapter. The second important factor is that the beams are not non-
deflecting. Since the distribution procedure is valid only for the
case of nondeflecting beams, this feature of the behavior of two-way
slabs requires special treatment. Consequently, throughout this bulle-
tin the beams are considered to be nondeflecting, and the effects of
deflection are left for future consideration.
The calculations described in the following sections had as their
object the determination of the effects of several variables on the
moments in panels of continuous slabs. The variables studied are
listed and discussed briefly below.
(a) Location of Panel.-The panels of a continuous slab may be
classified on the basis of location in three categories: interior, edge,
and corner panels. In order that the slabs to be analyzed should
have panels typical of all three types, a structure consisting of
twenty-five equal panels arranged in five rows of five panels each
was chosen. In such a structure the center panel is two rows distant
from an edge and should be fairly representative of a typical interior
, See for example H. M. Westergaard. "Formulas for the Design of Rectangular Floor Slabs
and the Supporting Girders," Proc. ACI, Vol. XXII, 1926, pp. 26-43.
ILLINOIS ENGINEERING EXPERIMENT STATION
panel. Similarly, a panel at the middle of a side of the slab is two
panels distant from a corner and may be considered as a typical
edge panel. Data may also be obtained for panels only one row re-
moved from an edge or corner.
(b) Ratio of Sides, b a.-Under similar conditions of loading
and restraint the moments in a given panel are dependent to a large
extent on the ratio of sides, b a. Three values of this variable were
considered: 0.5, 0.8, and 1.0. For each value, the slab analyzed con-
sisted of twenty-five panels of equal size. The analyses for the inter-
mediate ratio, 0.8, were less extensive than for the other ratios.
(c) Torsional Stiffness of Beams.-The effect of torsional stiffness
of the supporting beams is to increase the restraint offered to rota-
tion of the edges of a panel. This in turn results in a decrease in both
positive and negative moments for an interior panel, but in an in-
crease in negative moment on the exterior edge of an edge or corner
panel. Three values of the ratio of beam torsional stiffness to slab
flexural stiffness were considered, one of them being zero. The basis
for choosing the particular values used is discussed in Section 19.
(d) Type of Loading.-A floor slab is subjected to two types of
loading: (1) dead load, which is a uniformly distributed load over
the entire floor area, and (2) live load, which may or may not be
uniformly distributed. Throughout this bulletin the live load is as-
sumed to be uniformly distributed over each panel,' but not all
panels are assumed to be loaded. Two types of partial loading were
considered in addition to the dead-load condition of all panels loaded.
They are described in Section 20.
(e) Variation in Size of Panel.-All of the preceding discussion
has referred to slabs in which all panels are of the same size and
shape. It is also necessary to investigate the slab in which the adja-
cent continuous panels have different spans and different values of
b/a. One such slab was analyzed. It consisted of sixteen panels having
two different b/a ratios and two different span lengths. One value of
beam torsional stiffness and one type of live load only were con-
sidered for this structure. Further details are given in Section 21.
19. Torsional Rigidity of Beams
Two problems are considered in this section: first, the means of
computing torsional rigidity of a beam, and second, the determina-
tion of typical values of the ratio of beam torsional stiffness to slab
flexural stiffness.
I A limited amount of data for slabs with concentrated loads is given in Appendix C.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
The resistance a beam offers to rotation about its axis, its tor-
sional stiffness, may be computed if the geometrical and physical
properties of the section are known, and if some assumption is made
regarding the distribution of twisting moment along the length of
the beam. The distribution assumed herein is that of a sine-wave
having zero ordinates at the ends of the beam. This distribution is
not unreasonable and results in the following simple approximate ex-
pression for the torsional stiffness, Tb, of a beam having a span b:1
ut2GJ
Tb b- (4)
wherein
G = modulus of elasticity in shear of the material in the beam,
J = measure of torsional rigidity of the cross-section of the beam.
The quantity J for a rectangular section having a width v and
depth d is given by Timoshenko2 as:
v'd
J -= f. (5)
3
The notation of Equation (5) has been changed from that used by
Timoshenko. The quantity ft is a function of v/d and is tabulated
by Timoshenko; for the usual range of dimensions of reinforced
concrete beams, it may be expressed as:
f = 1 - 0.63-. (6)
d
The stiffness of a T-beam section is given by Nylander3 as
J' = J + Pt (0.33u - 0.17v - 0.211) (7)
wherein
J' = torsional stiffness of T-beam section
J = torsional stiffness of rectangular beam having dimensions
v by d
d = over-all depth of T-beam
v = width of stem of T-beam
u = width of flange of T-beam
t = thickness of flange.
In the analysis of continuous slabs, it is the ratio of beam tor-
sional stiffness to slab flexural stiffness that is of interest. Consider
N. \M. Newmark, "A Distribution Procedure for the Analysis of Slabs Continuous over
Flexible Beams," Univ. of Ill. Eng. Exp. Sta. Bul. 304 (1938); see pp. 28-20.
S. Timoshenko, "Theory of Elasticity." McGraw-Hill, New York, 1934. Equation (5) is
taken from the author's Eq. (157) on p. 249. Values of /i are tabulated as ki on p. 248.
3 Henrik Nylander. "Torsion and Torsional Restraint of Concrete Structures" (in Swedish).
Meddelandlen, Statens Kommnitt6 for Byggnadsforskning, Nr. 3, 1945, Stockholm. See Table 1(c),
p. 124.
ILLINOIS ENGINEERING EXPERIMENT STATION
a slab panel having dimensions b by a. What is desired is the ratio
Tb/Kb or Ta/Ka, where Tb and Ta are the torsional stiffnesses for the
beams of span b and a respectively, and Kb and K,, are the flexural
stiffnesses for the slab on edges b and a respectively. Considering
only the beam of span b, and assuming a rectangular beam section.
the stiffnesses are
T2G v3d v
Tb = 1 -0.63 (8)
and
kbN kbEt3
Kb - = (9)
b 12b
The stiffness ratio may then be expressed as
Tb 4 7r2 G d v3 (10)
-.= . .-- .. 1 - 0.63- . (10)
Kb kb E b t3 d
A similar expression may be obtained for Ta/Ka by substituting
a for b in Eq. (10), adding the term b/a to account for the lack of
reciprocity in the definition of Kb and Ka, and substituting the values
of v and d for the beam with span a.
In Eq. (10) the gross cross-section has been considered in com-
puting the stiffness of both the slab and the beam in accordance
with the usual practice for reinforced concrete. There may be some
question, however, as to the reliability of such a procedure in the
case of torsional stiffness. Another question may be raised: whether
the T-beam section should be used rather than a rectangular section
as is done in Eq. (10). Calculations made for a few of the typical
panels mentioned subsequently, and based on a flange width, u, of
one-fourth the span, indicated that the value of J' for a T-beam
section was only 20-40 percent greater than the value of J for the
corresponding rectangular section. The possible errors from using
the gross section and from using the rectangular section are of oppo-
site sign; consequently, it was decided to use Eq. (10) for the calcu-
lation of the stiffness ratios.
The next step was to determine, if possible, the values of T/K
for typical designs of two-way building slabs. The two sources of
such designs were:
(a) "Proposed Manual of Standard Practice for Detailing Rein-
forced Concrete Structures," published by the American Concrete
Institute, Detroit, 1946, and referred to herein as ACI Manual.
See Drawing 22 for two-way slab, on page 41.
(b) "Cost Estimates of Reinforced Concrete Floors," published
by Portland Cement Association, Chicago, 1940, and referred to
herein as PCA Cost Estimates.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
The ACI Manual contained typical drawings for a two-way slab
floor of 21 panels, having values of b a ranging from 0.77 to 1.0,
and span lengths of 17.50 to 22.75 ft. Values of T/K were computed
by means of Eq. (10) for 8 typical edge beams and 18 typical interior
beams. The average value thus obtained was 1.51, the minimum
1.21, and the maximum 1.87. There were no significant differences
noted for edge and interior beams, and only a slight effect of either
span length or b a was observed for the range of values considered.
The data in PCA Cost Estimates consisted of typical designs
for two-way slab floors having square panels with spans of 15, 20,
or 25 ft and designed for uniform live loads of 100 or 150 lb per
sq ft. Slab thicknesses varied from 4 to 7.5 in. Data were thus avail-
able for six different floors. Values of T/K were computed as before
by means of Eq. (10). The average value of the ratio was 1.77, the
minimum 1.18, and the maximum 2.28. The average ratio for de-
signs with 150-lb live load was about 30 percent greater than for
designs with 100-lb live load. A definite tendency was noted for
the value of T: K to decrease with increasing span length. The
average for the 25-ft spans was about 72 percent of that for the
15-ft spans.
The range of T K for all of the panels considered was from 1.21
to 2.29, with an average of about 1.60. The use of a T-beam section
would have increased these values by 20 to 40 percent. On the other
hand, the stiffness of the beam cross-section may not be as great as
that assumed, due to the effects of cracking. On the basis of the above
results it does not seem unreasonable to assume that the stiffness
ratio T K will normally lie in the range 1.5-2.0, and that it will
seldom if ever fall below 1.0.
20. Types of Loading
Three types of loading were considered in the analyses described
in this chapter. They are listed and discussed briefly below.
(a) Uniform load over entire area of slab, referred to herein as
"uniform loading." The dead load, or weight of the structure itself,
is of this type; and in certain types of storage buildings the live load
may correspond more closely to this type of loading than to either
of the other types mentioned below. In the design procedures for
flat slab floors specified in both the Joint Committee Report' and
the ACI Building Code2 the assumption of a uniform loading is
I "Recommended Practice and Standard Specifications for Concrete and Reinforced Con-
crete," Report of the Joint Committee on Standard Specifications for Concrete and Reinforced
Concrete. Published by American Concrete Institute, and others. June 1940.
.."Building Regulations for Reinforced Concrete (ACI 318-47)," American Concrete Institute,
Detroit, 1941. (See s-peýially S.tlion 1003.)
58 ILLINOIS ENGINEERING EXPERIMENT STATION
(a) (b) Cc)
C
A
F
V
G
B
H
A/
7
Maxmu Poit,-e Mmet
Notation 1 ax/mum Posit/ie M- omen i//
Zoawded Paane/s, and llaximum
Negaft/4e Afomen, at Exterior
Eda'e of Loaded Panes for-
T/Kfo a
(09) e) (f)
7-
EA S5 HI CE a 8H FB 5 GZ
Maximum Neegaive Mfoments on Inter/or Edyes as /iN/4oaed
FIG. 18. TYPICAL CHECKERBOARD LOADINGS
implied by the nature of the expression used for total moment.
(b) Loading for maximum moments, referred to herein as
"checkerboard loading." In this type, the load is considered as uni-
formly distributed over each panel, but only those panels are loaded
which contribute positively to the moment being considered. Typical
patterns of loading are illustrated in Fig. 18. For positive moment
in the interior of a panel, alternate panels are loaded in checkerboard
fashion. The same pattern is used to produce maximum negative
moment at the exterior edge of an edge panel, when torsional re-
'//11 In i
CP At0/1
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
straint of the beams is present. For maximum negative moment over
an interior beam, the checkerboard pattern is applied on either side
of, and symmetrical about, a line passing along the edge in question.
Such loadings are illustrated by the lower sketches in Fig. 18.
Checkerboard loading has been used by Westergaard for the calcu-
lation of both positive and negative moments in two-way slabs.1 It
is recommended for positive moments in both the Joint Committee
Report, Sec. 803(a)(1), and the ACI Building Code, Sec. 702(a)(2).
It is also specified in the ACI Code, Sec. 1002(a)(6), for the design
of flat slabs as continuous frames.
(c) The third type of loading is designated herein as "single-panel
loading." For positive moment, or for negative moment at an ex-
terior edge, only the panel in question is loaded. For negative mo-
ment over an interior beam, only the two adjacent panels are loaded.
The only precedent for this type of loading for positive moment is
the Joint Committee Report, Sec. 806(b)(2), with reference to mo-
ments in one-way slabs. Its use for negative moments, however, is
recommended throughout both the Joint Committee Report, sections
803(a)(2) and 806(b)(1), and the ACI Code, sections 702(a)(2)
and 1002(a)(6).
Three possible arrangements of live load which may be considered
in the design of two-way slabs have been presented. In order of
severity, as measured by the maximum moments produced, they
rank as follows: 1) checkerboard, 2) single-panel, and 3) uniform.
Some consideration also should be given to the relative probabilities
of occurrence of each of these patterns. Obviously, the probability
of obtaining a "single-panel loading" is very high, almost certain,
since only one or two panels need be loaded. Second in order comes
the uniform loading which is quite likely to occur in warehouses or
similar structures if the aisle space is small or if the aisles are several
panels apart. Even with wide or closely spaced aisles, it is possible
to have several panels in the same row or in alternate rows fully
loaded. The least probable of all the loading patterns considered is
the checkerboard loading, which is necessary to produce absolute
maximum moments. While a partial checkerboard loading might
occur occasionally, the probability of exact duplication of the neces-
sary pattern would seem to be small.
Data are given in Section 22 which permit a comparison of the
moments produced by each of the three types of loading described
in this section.
' H. M. Westergaard, "Formulas for the Design of Rectangular Floor Slabs and the Sup-
porting Girders," Proc. ACI, Vol. XXII, 1926, pp. 26-43.
ILLINOIS ENGINEERING EXPERIMENT STATION
21. Outline of Analyses
Two types of structures are considered in this chapter: slabs
with equal panels and slabs with unequal panels. The calculations
for moments in each type are outlined and described briefly in the
following paragraphs.
The major portion of this investigation was devoted to slabs
having twenty-five equal panels arranged in five rows of five panels
each. Three variables were considered: 1) the ratio of sides, b a;
2) the ratio of beam torsional stiffness to slab flexural stiffness, T K;
and 3) the type of loading. Three b/a ratios were considered: 0.5,
0.8, and 1.0. Plans of slabs having each of these ratios are given in
Fig. 19a, b, and c respectively. The values 0.5 and 1.0 represent the
limits of applicability of the distribution procedure. The intermediate
value, 0.8, was used for a limited number of analyses in order to
establish the nature of the variation of the most significant quantities
between the limiting values of b/a = 0.5 and 1.0.
Three values of T/K were considered: 0, 1, and 2. The zero value
applies to slabs supported on bare steel beams and to other cases
in which the beams provide no torsional restraint. The value of
T K = 1 was considered a minimum probable value for this ratio
in an actual two-way slab supported on beams cast monolithically
with the slab. Since all interior moments are decreased as the value
of T K is increased a conservative design procedure should be based
on a reasonable minimum for this ratio. The value of T/ K= 2 was
chosen as a maximum. This case must be considered, since the edge
moment at a discontinuous edge exists only as a result of torsional
restraint offered by the edge beam, and thus increases as T K
is increased.
The three types of loading described in Section 20 were used in
these analyses. The particular loadings used for each of the slabs
analyzed are indicated in Table 9. It may be noted from this table
that only eight different slabs were analyzed; a value of T/K=0
was not considered for b/a = 0.8. All three types of loading were
considered for the slabs with b a = 0.5 and 1.0, while only the uniform
and single-panel loadings were used for the structure with b /a=0.8.
Since the case of T K=2 was of interest only for moments at a
discontinuous edge, the live-load types of loading were usually ap-
plied only for the calculation of such moments for this value of T K.
In all the slabs having equal panels, the length, b, of the short
span was taken as 10 ft, and the uniform load, w, for the loaded
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
r- -tc-b/o.
Scr
TC ¢F B
I
£e D A-
A G I
(a))- b/= O.6
c
E
A
F
D
a
8
H
I
(c)- b/a =1/.0
E
A
G
H
(b )- b/a =0.e
NorT: Uniform load of
w /OO /b. per sq'. 4f.
over enf7re area of each
loadead pfanel.
4 C/ EZ E/b Cz
,'& /O
T E/a 12
VI -
(d)- /neqwua/ Pane/s
FIG. 19. PLANS OF SLABS ANALYZED
E3 I/ I3
CS
rh-Al/,
I
r
rý -
-b=/0
ILLINOIS ENGINEERING EXPERIMENT STATION
panels was assumed to be 100 lb per sq ft. These values were so
chosen as to result in convenient units for the average moments.
All moments were computed by means of the distribution pro-
cedure described in Chapter II, using the constants given in Table 1.
The method of calculation finally adopted consisted of loading one
panel at a time and computing the resulting moments on the edges
of all panels. Edge moments for a particular combination of loaded
panels were then obtained by superposition, and positive moments
in the interior of the panels were computed from these values of the
TABLE 9
SUMMARY OF TYPES OF LOADING CONSIDERED IN ANALYSES OF SLABS
HAVING TWENTY-FIVE EQUAL PANELS
Values of b/a
T
K
0.5 0.8 1.0
Uniform Uniform
0 Checkerboard Checkerboard
Single-panel Single-panel
Uniform Uniform Uniform
1 Checkerboard Checkerboard
Single-panel Single-panel Single-panel
Uniform Uniform Uniform
2 Checkerboard* Checkerboard
Single-panel* Single-panel* Single-panel
* Loaded only for maximum edge moment at an exterior edge and maximum positive moments
in edge panels.
edge moments. In practically all cases, however, the moments for
all panels loaded were obtained by direct calculation rather than by
superposition. The direct procedure was also followed for the checker-
board loadings on the slabs having T K = 0. These results were later
used as a check on the superposition procedure employed for the
single-panel loadings on the same structures.
For the structures having T K = 0, the exterior edges were simply-
supported and use was made of the modified distribution constants
in order to simplify the calculations. Modified constants were used
likewise in those cases for which the loading was symmetrical.
Advantage was also taken of symmetry in reducing the number of
panels for which the moments had to be computed. For example,
in the slabs having b a =0.5 or 0.8, moments were computed only
for the panels designated by the letters in Fig. 19a and b. For the
slab with b a= 1.0, moments were computed only for the panels
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
labeled C, F, B, D, H, and I in Fig. 19c. A total of 72 different analyses
were made: 33 for b a=0.5, 28 for b/a=1.0, and 11 for b/a=0.8.
The one slab having unequal panels is illustrated in Fig. 19d. It
consists of 16 panels of three different sizes or shapes arranged in
such a way that each type of panel occupies a corner, edge, and
interior position. Two ratios of sides were used, b/a=0.8 and 1.0.
The square slabs were of two sizes, 10 ft and 12.5 ft on ., side re-
spectively. The three shapes are designated as follows: 1) rectangular
panel, 2) large square panel, and 3) small square panel. The letter
designation indicates the position of the panel-corner, edge, or
interior. In Fig. 19d it may be noted that there are two edge positions
for the rectangular slab: one with the short side on the edge (Elb),
and the other with the long side on the edge (Ela).
The object in analyzing the slab with unequal panels was to
determine how the maximum moments in a given panel were affected
by variations in the size and shape of the adjoining panels. It was
deemed sufficient for this purpose to consider only T/K = 1, and to
use only the uniform and single-panel loadings. As in the case of
the other slabs, a uniform load, w, of 100 lb per sq ft was used.
All moment calculations were made by the distribution procedure.
Throughout all the analyses discussed in this section, the moments
computed were the average moments on the section in question. The
results for both types of slabs are presented and discussed in the
following sections.
22. Results of Analyses for Slabs with Equal Panels
The moments obtained from the analyses of slabs having twenty-
five panels are given in Tables 10, 11, and 12. The values tabulated
are average moments in ft-lb per foot of width, or simply pounds.
Plans of the slabs analyzed are shown in Fig. 19. The moments in
each table are grouped by types as follows: negative moments over
the interior beams, negative moments at an exterior edge, and posi-
tive moments. Moments in the short and long spans of the rectangular
panels are listed separately. The edge of a panel is designated by
two letters corresponding to the designations for the two adjoining
panels. For example, CF refers to the edge between panels C and F.
In each category the panels or edges are arranged with the corner
panel first, then the edge panels, and finally the interior panels.
In this section the effects of three important variables are
considered: 1) location of panel with respect to an edge, 2) type
of loading, and 3) value of T K, the relative torsional stiffness
of the beams.
64 ILLINOIS ENGINEERING EXPERIMENT STATION
Effect of Location of Panels.-Several important conclusions may
be drawn from the data in Tables 10, 11, and 12 concerning the
variation of moment for panels at various locations relative to the
edge of the slab. The general trend is for the moments in a panel
to increase as the restraints at one or more edges are diminished.
Thus the smallest moments are usually found for the interior panels;
the next largest for an edge panel, with one discontinuous edge; tile
greatest for a corner panel, with two discontinuous edges.
The greatest difference in moments for the various panels is
found for uniform loading on a slab having T K= 0; that is, no
TABLE 10
'MOMENTS IN CONTINUOUS SLABS HAVING TWENTY-FIVE EQUAL PANELS: b/a =0.5
See Fig. 19a for plan of slab and system of designating panels. All values are average moments
in panels for a uniform load w of 100 lb per sq ft.
T/K=0 T/K=1 , T/K=2
LOADING* UL SP CB IUL SP CB I. SP 1 CB
(1) (2) (3) (4) (5) (6) (7) i (8) (9) ( 10) (11) (12)
Negative Short CF 716 815 846 644 687 1 692 617
Moment Span FB 586 760 794 577 665 678 572
over ED 639 771 806 612 677 685 597
Interior AG 640 772 810 612 676 683 598
Beams DH 545 720 781 554 655 674 554
GI 544 720 782 554 655 676 556
Long CE 536 683 725 412 483 494 376
Span EA 517 668 719 407 482 495 376
FD 277 549 594 305 447 456 311
BH 328 546 622 318 450 474 316
DG 283 539 599 306 445 460 310
1 HI 324 537 623 317 449 473 317
Negative Short C 0 0 0 298 346 354 386 429 435
Moment Span E 0 0 0 282 339 352 375 425 433
at A 0 0 0 282 339 352 374 425 433
Exterior
Edge Long C 0 0 0 207 246 254 251 285 290
Span F | 0 0 0 152 228 237 208 275 281
B 0 0 0 159 229 243 211 275 283
Positive Short C 393 485 522 324 360 368 306 330 333
Moment Span F 286 423 469 289 348 353 284 329
P 310 420 480 292 347 362 286 324 330
E 333 445 492 304 353 361 296 331
A 336 444 493 305 353 363 295 327 331
D 272 393 461 276 341 357 277 330
G 271 393 462 277 341 358 277 329
H 281 391 466 278 341 359 279 331
I 282 391 467 278 341 360 279 322 330
Long C 115 132 139 102 108 110 98 103 103
Span F 92 120 129 94 106 107 93 103
B 98 119 131 95 106 108 94 102 103
E 106 126 135 99 108 109 96 103
A 106 126 135 99 108 110 96 102 103
D 91 116 127 92 105 108 93 103
G 91 116 127 92 105 108 93 103
H 94 115 129 93 105 108 93 103
I 94 115 129 93 105 1 108 92 102 102
* -L= Uniform loading over all panels.
SP = Single-panel loading.
CB = Checkerboard loading.
MOMENTS IN TWO-WAY CONCRETE FLOOR SL.AH
torsional stiffness of the beams. Even for this case, however, there
was no significant difference between the positive moments in any
of the interior panels, whether they were one or two rows distant
from the edge of the slab. In a square panel, or in the long span of
a rectangular panel, the positive moments in an edge panel on a
section parallel to a discontinuous edge were no different than those
for an interior panel. However, the moments on a section perpen-
TABLE 11
MOMENTS AND COMPARISONS FOR CONTINUOUS SLABS HAVING
TWENTY-FIVE EQUAL PANELS: b/o =0.8
See Fig. 19b for plan of slab and system of designating panels. All moments are average
moments in pounds for u' = 100 lb per sq ft.
LOADING*
(1) (2)
Negative Short
Moment Span
over
Interior
Beams
Long
Span
Negative Short
Moment Span
at
Exterior
Edge Long
Span
Positive Short
Moment Span
Long
Span
(3)
CF
FB
ED)
AG
DH
GI
CE
EA
FD
BH
DG
HI
C
E
A
C
F
B
C
F
B
E
A
D
G
H
I
C
F
B
E
A
1)
H
I
MOMENTS COMPARISONS
UL T/K=
SP /, T/K=1 °
T/K= 1 2 UL SP
T/K=1 T/K=2 (4) (6) (6) (7)
SPl( SP S ) (7) (4) (5)
(4) (5) (6) (7) (8) (9) (10) (11)
"449 487 429 92 95
426 478 414 89 97
404 473 401 85 99
404 473 401 85 99
389 465 389 84 100
389 465 389 84 100
376 426 353 88 94
371 422 351 88 95
309 401 310 77 100
311 401 311 78 100
309 399 310 77 100
311 399 311 78 100
218 248 279 305 88 91 128 123
195 240 259 300 81 86 133 125
195 240 259 300 81 86 133 125
180 219 236 261 82 90 131 119
154 206 206 253 75 81 134 123
156 206 207 253 76 82 133 123
190 211 182 195 90 1 93 96 92
181 207 177 87 98
181 207 177 192 87 92 98 93
172 204 171 84 99
173 204 171 192 85 89 99 94
167 200 168 83 101
167 200 168 83 101
168 200 168 i 84 100
168 200 168 84 100
117 129 111 120 91 92 95 93
100 123 100 81 100
100 123 101 117 81 86 101 95
117 128 112 92 96
117 128 112 120 92 93 96 94
102 123 102 83 100
102 123 102 83 100
102 123 102 83 100
102 123 102 83 100
* See note at bottom of Table 10.
66 ILLINOIS ENGINEERING EXPERIMENT STATION
dicular to the discontinuous edge were greater than those for an
interior panel. This may be explained by the fact that a span with
restrained edges at both ends is stiffer than a span with one simply-
supported end. Consequently, a greater proportion of the load is
carried to the supports by the stiffer span, and the moments on
sections normal to the simply-supported edge are thus increased. In
the short span of a rectangular edge panel, the positive moments
are not greatly different from those in an interior panel. However,
the differences are just the opposite of those for the long span; that
is, the moments on a section parallel to a discontinuous edge are in-
creased more than those on a section perpendicular to such an edge.
Negative moments in a square panel or in the long span of a
rectangular panel are practically the same over all interior beams,
except over a beam perpendicular to a discontinuous edge where
they are slightly higher. In the short span of a rectangular panel
this effect is reversed; the moment over a beam opposite a discon-
tinuous edge is appreciably higher than over an interior beam, while
little difference is noted for the moment over a beam perpendicular
to an edge.
TABLE 12
MOMENTS IN CONTINUOUS SLABS HAVING TWENTY-FIVE EQUAL PANELS: b/a=1.0
See Fig. 19c for plan of slab and system of designating panels. All values are average moments
in pounds for uniform load w of 100 lb per sq ft.
LOADING*
(1) (2) . (3)
Negative CF
Moment FB
over FD
Interior BH
Beams DH
HI
Negative C
Moment F
at B
Exterior
Edge
Positive C
Moment F it
F II
B ±L
B II
D
H
I
T/K = 0
UL
(4)
401
376
291
290
293
290
0
0
0
156
154
106
154
109
119
118
120
SP
(5)
475
455
410
409
399
398
0
0
0
186
181
157
181
157
156
156
156
CB
(6)
488
479
428
431
435
434
0
0
0
196
195
171
195
173
177
177
178
T/K =1
UL SP CB
(7) (8) (9)
339 377 378
337 372 379
291 357 363
290 357 364
291 355 366
290 356 371
170 194 197
145 184 190
145 184 191
134 I 146 148
138 147 150
114 140 142
137 147 150
113 140 142
120 140 145
120 140 146
120 140 146
T/K=2
UL
(10)
321
319
290
291
291
292
217
192
194
128
132
116
131
115
120
120
119
SP
(11)
347
346
339
339
338
338
235
229
229
136
137
133
137
133
134
134
134
CB
(12)
348
348
342
342
342
342
239
231
233
137
139
137
138
135
137
137
136
* See note at bottom of Table 10.
f (1) indicates moment perpendicular to edge of slab.
(I1) indicates moment parallel to edge of slab.
Moments in panel H are same in both directions.
I
* See note at bottom of Table 10.
t (.L) indicates moment perpendicular to edge of slab.
(!l) indicates moment parallel to edge of slab.
.Moments in panel H are same in both directions.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
The effects described in the preceding paragraphs apply qualita-
tively to all the analyses made. Their magnitude, however, was
affected by a number of factors. It has been mentioned that the
greatest differences were obtained for uniform loading and T/K =0.
The use of checkerboard loading, however, reduced the differences
appreciably, as may be observed by comparing columns (4) and (6)
in either Table 10 or Table 12. This effect is understandable, since
the loading of alternate panels tends to decrease the effects of con-
tinuity. The differences for single-panel loadings were intermediate to
those for checkerboard and uniform loadings, as would be expected.
If a value of T/K= 1, representative of a minimum amount of
torsional stiffness, is assumed, the differences between the panels at
various locations are reduced to an almost negligible amount for
both the checkerboard and single-panel loadings, although fairly
large differences still exist for the uniform loading. For T/K = 1, the
maximum difference between the moments in a corner panel and in
an interior panel is 5 percent for the checkerboard loading and 8
percent for the single-panel loading. For higher values of T/K these
differences are proportionately less; for T/K=2 they are about
one-half as great.
In summary, it may be concluded that the maximum moments
obtained by partial loading of either the checkerboard or single-
panel type are practically independent of the location of the panel
with respect to an edge if torsional beam stiffness corresponding to
T/K = 1 or more is considered.
Effect of Type of Loading.-The greatest moments were produced
by the checkerboard loading, next greatest by the single-panel
loading, and least by the uniform loading of all panels. In order to
make quantitative comparisons of the moments from the three types
of loading, the ratios in Tables 11, 13, and 14 have been computed. In
preparing these tables the single-panel loading was used as a base.
A comparison of considerable interest is that between the mo-
ments for the two types of partial loadings. Ratios of checkerboard to
single-panel moments are given in columns (4), (5), and (6) of Tables
13 and 14, for b/a =0.5 and 1.0 respectively. In all cases, the greatest
moment is given by the checkerboard loading. For T/K= 0, the dif-
ference is from 3 to 16 percent for negative moments and from 5 to 19
percent for positive moments. However, the assumption of torsional
restraint by the beams reduces considerably the differences between
the moments for the two types of loadings. For T/ K= 1, the maxi-
mum difference is 6 percent and the average only about 3 percent. For
ILLINOIS ENGINEERING EXPERIMENT STATION
T/K=2, the maximum is 3 percent. Thus for a relatively small
amount of beam torsional stiffness, the difference between moments
for the two types of partial loading becomes negligible.
Comparisons between the moment due to single-panel and uni-
form loadings are of interest because the uniform loading represents
dead load while the single-panel load is used for live load, and thus
the relation between dead- and live-load moment coefficients may
be studied. Data for the comparisons are given in columns (7), (8),
and (9) of Tables 13 and 14, and in columns (8) and (9) of Table 11.
TABLE 13
COMPARISONS OF MOMENTS IN CONTINUOUS SLABS HAVING
TWENTY-FIVE EQUAL PANELS: b/a=0.5
See Fig. 19a for plan of slab and system of designating panels. All tabulated values are ratios
of moments in Table 10, expressed in percent.
T/K =
Column Numbers from
Table 10
(2)
Short
Span
Long
Span
Short
Span
Long
Span
Short
Span
Long
Span
(3)
(4)
104
105
105
105
109
109
106
108
108
114
111
116
* See note at bottom of Table 10.
CuL_ J
CB*
SP %
0 1 2
(6) (9) (12)
(5) (8) (11)
T/K=0
T/K=1
T/K=2
T/K=1 °
0
(4)
(5)
(7)
(5) (6)
1
(7)
(8)
(8)
UL*
(4)
(10)
SP
(5)
(11)
CB
(6)
(9)
(12)
UL
(10)
(7)
(13)
SP
(11)
(8)
(14)
(1)
Negative
Moment
over
Interior
Beams
Negative
Moment
at
Exterior
Edge
Positive
Moment
102
105
103
105
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
Tabulated values are the ratios in percent of the uniform-loading
moments to the single-panel moments. In all cases the ratio is less
than 100 percent. The ratio of moments for uniform loading to those
for single-panel loading for T, K = 1 varies between 71 and 94 percent
for negative moments over interior beams, and between 67 and 92
percent for negative moments at an exterior edge. Corresponding
ratios for positive moments are 68 to 96 percent. In all cases the
percentage values increase as T/K is increased. In rectangular panels
the ratio is greater for moments in the long span than for moments
in the short span. Ratios for the short span are approximately the
same as those for a square panel. In general, the differences between
moments for the two types of loading were greatest for an interior
panel and least for a corner panel. With uniform loading, considerable
restraint is imposed at the interior edge of a panel as a result of
continuity with adjacent loaded panels. This effect is of course
greatest for interior panels and least for edge and corner panels.
With single-panel loadings, however, the difference in degree of
restraint at continuous and discontinuous edges is not so great since
the adjacent panels are unloaded.
TAouL 14
COMPARISONS OF MOMENTS IN CONTINOUs SLABS HAVING
TWENTY-FIVE EQUAL PANELS: b/a = 1.0
See Fig. 1.9c for plan of slab and system of designating panels. All tabulated values are ratios
of moments in Table 12 expressed in percent.
CB* UL T/K-0 T/K=2
SP SP T/K=- T/K-l1
T/K= 0 1 2 0 1 2 UL* SP CB UL SP CB
Column Numbers from (6) (9) (12) (4) (7) (10) (4) (5) (6) (10) (11) (12)
Table 12 (5) (8) (11) (5) (8) (11) (7) (8) (9) (7) (8) (9)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
Negative CF 103 100 100 84 90 93 118 126 129 95 92 92
Moment FB 105 102 101 83 91 92 112 122 126 95 93 92
over FD 104 102 101 71 81 86 100 115 118 100 95 94
Interior BH 106 102 101 71 81 86 100 115 119 100 95 94
Beams DH 109 103 101 73 82 86 101 112 119 100 95 94
HI 109 104 101 73 82 86 100 112 117 101 95 92
Negative C 101 102 88 92 128 121 121
Moment F 103 101 79 84 132 125 121
at B 104 102 79 85 134 125 122
Exterior
Edge
Positive C 105 101 101 84 92 94 116 127 133 95 93 92
Moment F _1* 108 102 102 85 94 96 112 123 130 96 93 93
F II 109 102 103 68 82 87 93 112 120 102 95 96
B AL 108 102 101 85 93 96 112 123 130 96 93 92
B I| 110 102 101 69 91 86 97 112 122 102 95 95
D 113 104 102 76 86 89 99 111 122 100 96 94
H ! 113 104 102 76 86 89 98 111 121 100 96 94
I I 114 104 102 77 86 89 100 111 122 99 96 93
* See Tables 10 and 12 for notation.
ILLINOIS ENGINEERING EXPERIMENT STATION
Effects of Beam Torsional Stiffness.-Several effects of variations
in T/K have been noted in the preceding paragraphs. Further com-
parisons are made possible by the ratios given in columns (10)
through (15) of Tables 13 and 14, and columns (10) and (11) of
Table 11. In each case the value of T/K=1 has been taken as a
base, and ratios of moment for other values of T/K to those for
this value are tabulated.
In general, all interior moments are greater for TiK = 0 than for
T/K= 1. A few exceptions to this rule are noted for uniform loading
where decreases were obtained at certain locations. The observed in-
creases were greatest for checkerboard loadings and least for uni-
form loadings. As would be expected the differences were greater for
corner or edge panels than for interior panels. In rectangular panels
the increase in negative moments for T/K=0 was greatest for the
long span, while for positive moments it was greatest for the short
span. Percentage increases for the square panels lay between those
for the long and short spans of the panels having b/a=0.5. The
increases in moments from T/K = 1 to T/K = 0 varied considerably
depending on the type of loading, b/a, etc. Average values, however,
were as follows: 4 percent for uniform loading, 17 percent for single-
panel loading, and 26 percent for checkerboard loading.
As the value of T/K was increased from 1 to 2, all interior mo-
ments were decreased and all negative moments at an exterior edge
were increased. The greatest decrease in the interior moments was
9 percent, and the average about 5 percent. This 5 percent decrease
from T/K= 1 to 2 may be compared with the decrease of about 18
percent from T/K = 0 to 1 for interior moments and partial loadings.
It is thus seen that by far the greatest proportion of the decrease
in interior moments due to the assumption of beam torsional stiffness
is obtained by assuming T/K = 1.
The increase in edge negative moments corresponding to a change
in T/K from 1 to 2 was greatest for uniform loading and about equal
for the other two loadings. For uniform loading the increase ranged
from 21 to 37 percent with an average of 32 percent; for either
checkerboard or single-panel loadings the range was 14 to 25 per-
cent, with an average of 22 percent. Obviously, even greater in-
creases would be obtained if higher values of T/K were considered.
For T/K equal to infinity the edge moments are those for a fixed-edge
panel. It is informative to note that for single-panel loadings the
edge moments for T/K = 1 averaged 65 percent of those for T/K =
infinity; for T/K = 2 the corresponding figure was 80 percent. It was
estimated that for this type of loading a moment equal to about 85
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
percent of the fixed-edge moment would be obtained for T/K=3.
This constitutes an increase of only about 5 percent over the moments
for T K= 2. Whereas a value of T/K = 1 constituted a conservative
assumption for the calculation of interior moments which are de-
creased by an increase in torsional stiffness, it is evident that a value
of T K not less than 2 should be used when computing the moments
at an exterior edge since they are increased as T/K increases.
Positive Moments in Interior Panels.-The interior panels of a
slab subjected to uniform loading are for all practical purposes fixed
on all four edges. This is easily verified by comparing the edge
moments for such panels with the fixed-edge moments in Table 1.
It is the positive moments in these panels which are of interest here.
Consider first the interior panel, I, of a slab having b/a= 1. The
positive moment due to uniform loading is obtained from column
(4), (7), or (10) of Table 12 as 120. This is the value computed by
means of the distribution procedure and the positive moment cor-
rection factors of Table 1. The correct value of the average positive
moment in a fixed-edge slab is determined from Section 31, Appen-
dix A, as 96. The approximate procedure thus gives a value for the
positive moment in a square panel about 25 percent too high.
Similarly, in an interior panel having b/a=0.5, the computed posi-
tive moment in the short span is 278 as compared to 255 for the
exact average moment in a fixed-edge slab. In this case the error is
only 9 percent. For the long span the approximate moment is 93
and the exact value is 51. The rather large discrepancy in this case
is of little importance, however, since the moments in question
are small.
The above differences between the approximate and exact posi-
tive moments for the few cases which permit comparisons to be
made are probably typical of all the results for partial loadings with
T/K = 1 or more. The errors for the other panels, however, are not
likely to exceed those given above for the interior panels. The source
of these errors is of course the conservatism of the positive moment
factors which has been discussed previously. Since a high degree of
accuracy could not be obtained, these factors were chosen so that
the larger errors would always be on the side of safety.
23. Results of Analyses for Slab with Unequal Panels
The slab analyzed consisted of sixteen panels of three different
sizes or shapes, as illustrated in Fig. 19d. Torsional stiffness of the
beams corresponding to a value of T/K = 1 was assumed throughout;
the value of K used was the average for the two adjacent panels.
ILLINOIS ENGINEERING EXPERIMENT STATION
The types of loading considered included only uniform loading over
all panels and single-panel loading. In the latter, one panel is loaded
to obtain maximum positive moments or maximum negative mo-
ments at an exterior edge, while two adjacent panels are loaded to
obtain maximum negative moments over an interior beam. Moments
TABLE 15
POSITIVE AND EDGE NEGATIVE MOMENTS FOR SLAB WITH
SIXTEEN UNEQUAL PANELS
See Fig. 19d for plan of slab and notation. All values are average moments in pounds for a uni-
form load w of 100 lb per sq ft. See text, Section 23, for description of calculations.
Single-Panel Loading
From Equal
Panels
(5)
211
230
207
228
140
123
140
204
219
146
129
218
123
228
147
200
140
128
219
146
-248
-184
-240
-194
-219
-288
-206
-303
Uniform Loading
By Direct From Equal
Distribution Panels
(6) (7)
190 190
212 216
178 181
213 209
112 114
99 102
120 120
174 173
191 187
133 134
114 117
185 178
95 100
213 209
138 138
166 167
120 120
115 117
191 187
133 134
-217 -218
-142 -145
-197 -195
-169 -170
-185 -180
-236 -226
-144 -156
-274 -266
* Axes indicated on Fig. 19d.
for each case were computed by means of the distribution procedure
described in Chapter II. The results are given in Table 15 and
Fig. 20.
Results are given in Table 15 only for positive moments and for
negative moments at an exterior edge. The values in columns (4)
and (6) are those determined by means of the distribution procedure
for the slab with unequal panels. The values in columns (5) and (7)
were obtained from the analyses of slabs having twenty-five equal
panels. For each of the unequal panels in the slab of Fig. 19d, a
Direction
of
Moment*
(2)
M.
M,
(1)
Positive
Moment
Negative
Moment
at
Exterior
Edge
Panel
(3)
Cl
E2
Elb
C2
E3
11
13
Ela
12
C3
Cl
E2
Elb
C2
E3
11
13
Ela
12
C3
Cl
E3
Ela
C3
Cl
E2
Elb
C2
By Direct
Distribution
(4)
209
230
206
229
140
123
140
203
219
146
128
218
123
229
147
200
140
128
219
146
-250
-185
-240
-194
-221
-286
-206
-302
MOMENTS IN T"WO-WAY CONCRETE FLOOR .SLABS
KE f Moment// for eiwua/ pane/s. See moment for
-487 -589' edge CF 11 Tab/le //, Col S and Table /2, Col 8.
- 34 + 34 Dislribu/e / difference.
-52/ -55 Nt/e moment for unequal pane/s.
(-523) (-552) Actual moment for enequal pane/s.
C/ E2 Elb C2
-487 -8.9 -6S/ -478 -478 -589
-34 +34 +34 -34 -37 +37
-52/ -555 -547 -5/2 -5/5 -552
(-S23) (-562) (-547) (-S/4) (-S/6) (-S53)
E3-357 -401 T1 -399 -356 13-35 -40/ Elb
- /5 + /15 + 14 - /4 - /I + /5
-372 -386 -38S -370 -372 -386
(-374) (-387) (-386) (-372) -373 (-388)
E112 1/ I E2
-473 -558 -556 -465 -473 -558
- 28 +28 + 30 -30 -28 +28
-SO/ -530 -526 -495 -50/ -SS0
(-503) (-5.29) (-525) (-493) (-495) (-530)
CS -377 -422 Ela -422 -37 E3-377 -426 C/
- /5 + I5 + 17 - 17 - /6 + /6
-392 -407 -40S -389 -393 -4/10
(-392) (-407) (-406) (-389) (-391) (-413)
Aore-: See Fig. /9 for dimensions of pane/s. A// moments
in pounds for w=OO /lb. per sq. ft. /n pane/s adja-
cent to edge consi/dered.
FIG. 20. NEGATIVE MOMENTS OVER INTERIOR BEAMS FOR SLAB WITH
SIXTEEN UNEQUAL PANELS; SINGLE-PANEL LOADING, AND T/K = 1
corresponding panel having the same b/a and T/K ratios, occupying
the same position relative to an edge or corner, and loaded in the
same manner was chosen from the slabs of Fig. 19a, b, or c. The
moments in this panel, corrected if necessary for differences in span
length, are those recorded in columns (5) and (7). For example,
panel C1 of Fig. 19d corresponds to panel C of Fig. 19b; thus the
positive moment of 211 in the first line of column (5), Table 15,
was obtained from column (5) of Table 11 on the line labeled "Posi-
tive Moment; Short Span; C." Similarly the positive moment of 230
ILLINOIS ENGINEERING EXPERIMENT STATION
in the second line of column (5), Table 15, is equal to (12.5/10)2
times the moment of 147 in column (8) of Table 12 on the line labeled
"Positive Moment; F; (1)." In this case a correction is necessary
to take account of the difference in span length.
The results given in Table 15 for the single-panel loading indicate
that the positive moments and the negative moments at an exterior
edge of a given panel are practically independent of the size and
shape of the adjacent panels. Although in this study the variation
in span length of adjacent panels was only 25 percent, the very
close agreement obtained suggests that much greater differences
could exist without seriously affecting the accuracy of the results.
For uniform loading, the agreement between moments for equal and
unequal panels is not as good as for the single-panel loading. The
maximum difference is about 8 percent, which is not excessive in
view of the nature of the approximation involved. Some caution
should be exercised, however, in extending this procedure to struc-
tures having differences in adjacent span lengths greater than about
25 percent if uniform loading is considered.
Negative moments over the interior beams for single-panel load-
ings are presented in Fig. 20. In this figure the moments applying to
each edge are written in the appropriate panel adjacent to that
edge. The values in the top line were obtained from the analyses
for slabs with equal panels in the same manner as that described in
connection with Table 15. For example, the moment of -487 at
the left in the "Key" was obtained from column (5), Table 11,
on the line labeled "Negative Moment; Short Span; CF." Similarly,
the moment of -589 at the right is equal to (12.5/10)2 times the
moment of -377 on the first line of column (8), Table 12.
As may be seen from the figure, the moments obtained from the
equal panel solutions are different on the two sides of an edge because
of the difference in size and shape of the adjacent panels. These
different moments may be "balanced," however, by using the dis-
tribution procedure in a manner similar to that used in balancing
fixed-edge moments. Since the slab stiffness K differs only slightly
for adjacent unequal panels,' the value of T used in the distribution
procedure was assumed equal to the average of the K values for
the adjacent panels. For balancing the equal panel moments, a
further simplification may be made by assuming the values of K
to be equal in adjacent panels. If T/K = 1, the correction moments
are thus one-third of the difference between the moments in adja-
cent panels. This fraction would be different for other values of
I See Fig. 4 and discussion in Section 6.
MO.M1ENTS IN TWO-WAY CONCRETE FLOOR SLABS
T/K. Moreover, for greater variations in adjacent span lengths
than those considered here or for slabs with panels having different
thicknesses, more accurate assumptions regarding the relative magni-
tudes of K for adjacent panels might have to be made. The correction
moments, obtained as described above, are written on the second
line of the calculations in Fig. 20.
The corrected equal-panel moments are written on the third line,
and may be compared with the moments obtained by means of the
distribution procedure for the slab with unequal panels, which are
written on the bottom line in parentheses. The agreement between
these two sets of moments is seen to be excellent. The maximum
difference is only about 1 percent, and it seems likely that slabs with
a considerably larger variation in adjacent span lengths could be
handled by means of the procedure used in this case.
Comparisons similar to those in Fig. 20 were also made for uni-
form loading. The agreement for this case was not as good as for
single-panel loading, but the maximum error did not exceed 6 percent.
This relation between the results for uniform and single-panel load-
ings is substantially the same as that obtained for positive moments
and for negative moments at an exterior edge.
Although the studies described in this section were not very ex-
tensive, it is believed that the following conclusions are justified.
(1) For slabs consisting of dissimilar panels having the same thick-
ness and not more than a 25 percent variation in adjacent span
lengths, the positive moments and the negative moments at an ex-
terior edge may be assumed equal to the corresponding moments
in typical panels of a slab made up entirely of equal panels. (2) For
the same conditions as those just stated, the negative moments over
interior beams may be obtained from the corresponding moments
in a group of equal panels, corrected as follows: For a given edge,
determine the negative moment for each abutting panel from the
equal-panel solutions; then, where these moments are different, dis-
tribute two-thirds of the difference equally to the two panels. If
the slab thickness is different in adjacent panels, the first conclusion
above regarding positive moments and negative moments at an
exterior edge is still valid. However, for the calculation of interior
negative moments, two-thirds of the difference should be distributed
to the two panels in proportion to their respective stiffness factors,
which in this case may differ appreciably because of the difference
in thickness. The principles stated above have application in ex-
tending a design procedure based on the results of equal-panel
analyses to the case of slabs consisting of dissimilar panels.
ILLINOIS ENGINEERING EXPERIMENT STATION
24. Summary
The distribution procedure for the analysis of continuous slabs
has been applied to the study of moments in two-way reinforced
concrete floors. The relative ease with which moments may be com-
puted by this procedure made it feasible to undertake a large num-
ber of analyses and to investigate the effects of several variables.
The following factors were studied: 1) location of panel with respect
to an edge of the slab; 2) torsional restraint offered by the supporting
beams; 3) various types of loading, including all panels loaded and
two types of partial loadings; 4) ratio of sides of panels, b/a, for
slabs in which all panels are similar; and 5) combinations of panels
of various span lengths and values of b a.
Eight slabs consisting of twenty-five equal panels and one slab
consisting of sixteen unequal panels were analyzed for various loading
conditions. A total of 83 different analyses were made. A summary
of the calculations for the slab with equal panels is given in Table 9.
The numerical values of the moments obtained from the analyses
are presented in Tables 10-15 and in Fig. 20.
The following conclusions are believed to be justified on the basis
of the data presented and discussed in this chapter:
1. The ratio, T/K, of the torsional stiffness of a beam to the
flexural stiffness of the slab for typical two-way slabs supported on
concrete beams was found to vary from 1.2 to 2.3, with an average of
about 1.5. A reasonable minimum value to cover most cases would
appear to be 1.0, neglecting both T-beam action and cracking of the
concrete, two effects which act in opposite directions.
2. For T/K equal to or greater than one, the moments produced
in a continuous slab as a result of either single-panel or checkerboard
loadings are practically the same for all panels, regardless of their
location with respect to a discontinuous edge. Moments at a discon-
tinuous edge are of course different. For these same conditions, the
moments due to uniform loading are different in corner or edge
panels but are the same for all interior panels. In an edge panel, the
greatest effects were usually observed for moments on a section
perpendicular to a discontinuous edge.
3. The greatest moments were produced by the checkerboard
loading, and the least by the uniform loading. For T/K equal to
one or more, there was little difference between the moments for
checkerboard and single-panel loadings. The moments for uniform
loading, however, were somewhat smaller, ranging from 67 to 96
percent of those for partial loadings.
MOMENTS IN TWO-WAY CONCHRETEI FLOOR SLABS
4. As the torsional stiffness of the beams is increased, both the
positive moments and the negative moments over interior beams are
decreased. An average decrease of 18 percent was produced by an in-
crease in T K from zero to one, and an additional decrease of only 5
percent was produced by a further increase from one to two. It thus
appears that a major portion of the effect of torsional stiffness on the
interior moments is accounted for by the assumption of T/K = 1.
5. As the torsional stiffness of the beams is increased, the negative
moments at an exterior edge are likewise increased. For an increase
in T K from one to two, this increase averaged 32 percent for uni-
form loading and 22 percent for the partial loadings. The resulting
edge moments were on the average about 80 percent of those for a
completely fixed edge. It was estimated that an increase of T/K
from two to three would increase the edge moments only about 5
percent more. Thus a value of T/K = 2 would appear to be a reason-
able upper limit for the beam torsional stiffness.
6. Average positive moments in interior panels and probably also
in other panels were shown to be as much as 25 percent higher than
the exact average moments. This difference was explained as the
result of the conservative values of positive moment correction
factors used with the distribution procedure.
7. For slabs consisting of a number of panels of unequal sizes
and different shapes, but having not more than 25 percent variation
in span length for adjacent panels, the positive moments and the
negative moments at exterior edges may be taken as equal to the
corresponding moments in panels of a slab consisting solely of equal
panels. For the same conditions, negative moments over interior
beams may also be obtained from the moments in slabs with equal
panels, but a correction must be made as described in Section 23.
Four recommendations regarding the values of the several varia-
ables that should be used in the development of a design procedure
for two-way concrete slabs supported on beams providing torsional
restraint are based on the conclusions in the preceding paragraphs:
(a) The single-panel type of loading should be used for the calcu-
lation of moments due to live load.
(b) Uniform loading of all panels should be used for the calcula-
tion of moments due to dead load.
(c) Positive moments in all panels and negative moments over
interior beams should be computed for an assumed torsional stiffness
of the beams corresponding to TiK = 1.
(d) Negative moments at an exterior discontinuous edge should
be computed for T K = 2.
ILLINOIS ENGINEERING EXPERIMENT STATION
V. SUMMARY
The. objective of the studies reported in this bulletin was to
obtain a better understanding of the behavior of two-way slabs.
The approach to this problem has been made in two steps: 1) the
development of an approximate moment-distribution procedure for
the calculation of moments in uniformly loaded plates continuous
in two directions over rigid beams; 2) the application of this pro-
cedure to the study of the effects of several important variables
on the moments in two-way slabs.
The distribution procedure which has been derived herein is
analogous to the Cross moment-distribution method for the analysis
of continuous beams and frames; that is, the procedure is one in
which fixed-edge moments are calculated, unbalanced moments are
distributed in proportion to the relative stiffnesses of the elements
of the structure, and portions of the distributed moments on each
edge of a panel are carried over to the other edges. The distribu-
tion procedure as applied herein to plates is approximate, however,
and in this respect differs importantly from the Cross method. The
moments considered are the average moments acting on the edge
of a panel and are functions of the ratio of sides of the panel as well
as the span length and loading. The stiffness factors are generally
similar to those of the Cross method, but in addition to being func-
tions of the span length and depth of the slab, they too are dependent
on the ratio of sides, as are also the carry-over factors. The effect
of edge moments on the positive moments in the interior of a simply-
supported slab is determined approximately by means of correction
factors which are functions only of the ratio of sides of the panel.
The distribution procedure is described in Chapter II, and its
development is explained in detail in Chapter III.
The distribution procedure was applied to the calculation of
moments in a number of continuous slabs. Several variables were
studied, including 1) the location of a panel with respect to a dis-
continuous edge; 2) torsional restraint offered by the supporting
beams; 3) various types of loading, including all panels loaded,
and two types of partial loading; 4) ratio of sides of panels for slabs
in which all panels were similar; and 5) combination of panels of
various span lengths and ratios of sides. Eight slabs consisting of
twenty-five equal panels arranged in five rows of five panels each
and one slab consisting of sixteen unequal panels were analyzed
for various loading conditions. A total of 83 different analyses were
made. From these studies, certain conclusions were reached regarding
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS 79
the effects of the several variables, and the types of loading and
the values of torsional beam stiffness which should be considered
in the development of a design procedure. The studies mentioned
above are described in Chapter IV, and the results obtained and
conclusions drawn are summarized in Section 24.
Supplementary data in the form of exact solutions for moments
in various types of slabs are presented in the three appendixes
following. The solutions in Appendix A for uniformly loaded single
rectangular plates, and those in Appendix B for uniformly loaded
continuous plates, were used extensively in the development of the
distribution procedure. Rectangular plates with concentrated loads
are considered briefly in Appendix C.
ILLINOIS ENGINEERING EXPERIMENT STATION
APPENDIX A
MOMENTS IN UNIFORMLY LOADED RECTANGULAR PLATES
WITH VARIOUS EDGE CONDITIONS
25. Introduction
Solutions are given in this Appendix for moments in uniformly
loaded rectangular plates supported on all four sides and having
various combinations of fixed and simply-supported edges. The
moments were computed by means of the so-called ordinary theory
of flexure for slabs, and are referred to herein as "exact" values.
Actually, all the solutions involved the use of infinite series, and
the correctness of the results is consequently dependent upon the
number of terms used in the calculations. In general, the accuracy
of the analyses was such that the moments given herein are believed
to be correct to within three or four units in the last significant
figure. Moments on the edges of a panel, which are computed directly
from the terms in the series, are likely to be more accurate than
this; moments in the interior, which are obtained by one or more
corrections to the simply-supported slab moments, are probably the
least accurate.
The various types of edge conditions considered include 1) simply-
supported plates, 2) plates with one edge fixed, 3) plates with two
opposite edges fixed, 4) plates with two adjacent edges fixed, 5) plates
with three edges fixed, and 6) plates with all four edges fixed. The
solutions for one or two edges fixed, and for three edges fixed with
the short edge simply-supported, were obtained by the writers. How-
ever, in several cases, the solutions obtained by other investigators
were extended, and moments were computed at additional points
in the structure. This computation usually consisted of using the
terms in the series for edge moments to determine the distribution
of those moments across an edge, and to obtain the correction
moments for the calculation of positive moments at points in the
interior of a panel.
So far as possible a uniform notation has been used throughout
this Appendix, and wherever necessary the notation used in solu-
tions from other sources has been changed to conform. The system of
coordinates is shown in Fig. 12. The designations M. and My refer
to moments per unit of width in the direction of x and y respectively,
acting on a section normal to the x or y axis respectively. The spans
a and b are in the direction of x and y respectively, and except where
otherwise noted, the dimension b refers to the shorter side.
All the solutions given herein are for a value of Poisson's ratio
MOMENTS IN TWO-WAY CONCRETE FLOOR -I.ABS
equal to zero. The moments at an edge are independent of this
ratio; the moments in the interior are not, but may be determined
for other values of Poisson's ratio by means of the equations given
in Section 3 of this bulletin.
Average moments are computed in almost every case considered.
Where the moment in question is expressed as a sine series, the
average was obtained directly by computing the total moment from
the terms in the series. Thus, if
M = -A, sin --
a
then a
Mtot = E- A,, for n odd only,
nir
and Mav is equal to Mtot divided by the width of the section being
considered. Where series terms were not available but where values
of the moment were known for points spaced at regular intervals
across the section, as was frequently the case for positive moments
in the interior of a panel, the average moment was computed from
these values by means of Simpson's one-third rule for obtaining areas.
A brief summary of the moments computed and tabulated for
the various cases considered is given in Table 16.
26. Simply-Supported Plates
The moments in uniformly loaded simply-supported plates were
obtained from the results given by Leitz' and Bittner.2 These are the
only sources known to the writers in which moments are given for a
large number of points in the interior of the plate.
Leitz has tabulated moments in plates having b/a=0.5, 0.8,
and 1.0 for points at the intersections of lines at x/a and y/b =0,
0.1, 0.2, 0.3, 0.4, and 0.5; and moments for b/a=0 at the inter-
sections of lines at x/b=0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0, and
2.0, and y/b=0, 0.1, 0.2, 0.3, 0.4, 0.5. Coefficients are given for
both Mx and M, at each point, for the twisting moment M., and
for the principal moments and their directions. The tabulated mo-
ments are for a value of Poisson's ratio equal to zero. In presenting
the moments herein, the notation used by Leitz has been converted
to that of Fig. 12. Values are also given by Leitz for moments in both
directions at the middle of the plate, for b/a = 0, 0.200, 0.250, 0.333,
0.400, 0.500, 0.571, 0.667, 0.800, and 1.0.
' H. Leitz, "Die Beruchinung der frei aufliegenden, rechteekigen Platten." Forsch. auf dem
Gebiete des Eisenbetons, Heft XXIII. Wilhelm Ernst u. Sohn, Berlin, 1914.
2E. Bittner, "Moinententafein und Einflussfluchen fiir kreuzweise bewehrte Eisenbeton-
platten." Julius Springer, Vienna, 1938.
ILLINOIS ENGINEERING EXPERIMENT STATION
The values tabulated by Bittner are coefficients for M- and My
at the middle of a simply-supported plate loaded with a load, P,
distributed over a centrally loaded rectangle having a width tG in
the direction of the short span, lz, and a length t, in the direction
of the long span, ly. Values of M. P and M, P are tabulated for
ly, l= 1.0, 1.1, 1.2, 1.3, 1.4, and 1.5, for various values of t, l1 and
tz 'I. Because of certain reciprocal relations, the tabulated value of
M/P for given values of tx and t, is numerically equal to the quantity
M'/4xyw, where M' is the moment at point x, y in a simply-supported
TABLE 16
SUMMARY OF MOMENTS GIVEN IN APPENDIX A FOR UNIFORMLY
LOADED RECTANGULAR PLATES
Section Edge Values of Moments Given
Section Condition b/a
26 Simply- 0.5 Moments in interior of panel at Hio-points in
Supported 0.5 both directions
Plates 1.0
0.667 Moments in interior of panel at io-points in
0.714 short span and at various intervals in long span
0.770
0.800
0.833
0.909
27 One 0.500 Maximum and average moments on fixed edge
Short 0.625 and at midspan of both spans for all b/a. Average
Edge 0.714 moments on certain sections in interior for b/a=
Fixed 0.833 0.5 and 1.0, and maximum value of average posi-
0.909 tive moment in both spans
1.000
One 0.5 Same as above
Long 0.6
Edge I 0.7
Fixed 0.8
0.9
1.0
28 Same as 27 except for two opposite edges fixed
29 Two 0.5 Moments at iit-points on fixed edges. Average
Adjacent 1.0 moments on various sections in interior, and
Edges maximum value of average positive moment in
Fixed I both spans
30 One Short 0.50 Moments at !'2-points on fixed edges. Average
Edge 0.75 moments on various sections in interior, and
Simply- maximum value of average positive moment in
Supported I both spans
One Long 0.33 Moments at 3i-points on short edges and at ý12-
Edge 0.50 points on long edges. Average moments on vari-
Simply- 0.75 ous sections in interior, and maximum value of
Supported 1.00 average positive moment in short span, for b/a =
0.5 and 1.0
31 Four 0.500 Moments in both directions at Mo-points on
Edges 1.000 edges and in interior of panel
Fixed
0.667 Moments at to-points on edges only
0.571 Maximum and average moments on edges only
0.800
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
tx
plate loaded with a uniformly distributed load, w, when x =
1t 2
and y = --. Using this relation, it is possible to compute the mo-
2
ments at a number of points in the interior of the plates, limited
only by the range of tabulated values of t,, lx and tI 1,. In the nota-
tion of this bulletin, values of Mx and M, may be determined at
y b=0.1, 0.2, 0.25, 0.3, 0.4, and 0.5, and at various values of x/a
depending on the particular value of b 'a considered. In general,
data were available at sufficient values of 'a to permit easy inter-
polation for moments at the one-tenth points of span a.
The most significant moments obtained from the solutions by
Leitz and Bittner are given in Tables 17 and 18. Moments are given
in Table 17 for the one-tenth points across the width of the plate
at midspan. For the moment in the long span, M,, values are also
given for a section at or near that for which this moment is a maxi-
mum. For b 'a greater than 0.75, the maximum moment in the long
span occurs at midspan; for smaller values of b/a the location of
the section for maximulm moment is indicated in Table 18. The
maximum value of the moment in the short span, My, occurs at
midspan for all values of b/a. The average moments on the sections
considered are given in the last column of Table 17.
Maximum and average moments for the various value of b/a
are summarized in Table 18. All values for M, are for a section at
midspan. For the moment, M., in the long span, the average moment
is given both for the section at midspan and for the section on which
this moment is greatest.
The moments given by Leitz and Bittner for a plate having
b/a = 1.0 were never different by more than one in the last place.
27. Plates with One Edge Fixed
Moments are given in this section for uniformly loaded rectangular
plates having one edge fixed and the other three edges simply sup-
ported. Moments were computed at the fixed edge and in the interior
of plates having either a short or a long edge fixed. For a long edge
fixed, solutions were obtained for values of b/a =0.5, 0.6, 0.7, 0.8,
0.9, and 1.0; for a short edge fixed, the calculations were made for
b, a= 0.500, 0.555, 0.625, 0.714, 0.833, 0.909, and 1.000.
All solutions were obtained using the procedure developed by
N. M. Newmark and described in Bulletin 304.1 This procedure in-
volves the use of infinite series, and the degree of accuracy of the
' N. M. Newmark, "A Distribution Procedure for the Analysis of Slabs Continuous over
Flexible Beams," Univ. of Ill. Eng. Exp. Sta. Bul. 304. 1938.
ILLINOIS ENGINEERING EXPERIMENT STATION
results is dependent on the number of terms in the series considered.
The number of terms used herein for each of the plates studied were
as follows: For a long edge fixed, for b/a=0.5, 0.6, and 0.7-eleven
terms; for a long edge fixed for b/a =0.8, 0.9, and 1.0-nine terms;
for a short edge fixed for all values of b/a-seven terms. By eleven
terms is meant that the terms in the series through n =11 were
used, even though in this case the even-numbered terms are equal
to zero. The number of terms used in each case was so chosen as to
obtain moments which would not be in error by more than about
two in the fourth decimal place. This degree of accuracy applies
to the moments on a fixed edge and also to the moments in the
interior produced by the edge moments. These latter will be referred
TABLE 17
MOMENTS IN UNIFORMLY LOADED SIMPLY-SUPPORTED RECTANGULAR PLATES
Poisson's Ratio= 0. See Fig. 12 for notation.
M,/wb2, in short span, at x/a=
b y
a b
0.1 0.2 0.25 0.3 0.4 0.5 Av
0.500 0.5 0.0350 0.0631 ...... 0.0825 0.0934 0.0964 0.0651
0.667 0.5 0.0241 0.0449 0.0532 0.0605 0.0698 0.0727 0.0475
0.714 0.5 0.0218 0.0409 0.0485 0.0549 0.0637 0.0665 0.0433
0.770 0.5 0.0193 0.0362 0.0432 0.0490 0.0569 0.0595 0.0385
0.800 0.5 0.0180 0.0341 ...... 0.0460 0.0535 0.0560 0.0362
0.833 0.5 0.0167 0.0315 0.0377 0.0430 0.0500 0.0523 0.0338
0.909 0.5 0.0142 0.0267 0.0321 0.0367 0.0427 0.0447 0.0288
1.000 0.5 0.0115 0.0220 0.0263 0.0299 0.0351 0.0368 0.0236
Mz/wb5, in long span, at y/b=
b x
a -a
0.1 0.2 0.25 0.3 0.4 0.5 Av
0 (x/b= 0.0077 0.0143 ...... 0.0193 0.0227 0.0232 0.0152
0.3)
0.500 0.5 0.0054 0.0103 ...... 0.0141 0.0166 0.0174 0.0111
0.2 0.0079 0.0149 ...... 0.0203 0.0236 0.0248 0.0160
0.667 0.5 0.0087 0.0165 0.0199 0.0227 0.0266 0.0280 0.0179
0.3 0.0091 0.0172 0.0206 0.0236 0.0275 0.0289 0.0185
0.714 0.5 0.0094 0.0179 0.0215 0.0246 0.0288 0.0302 0.0193
0.357 0.0095 0.0181 0.0217 0.0248 0.0291 0.0305 0.0195
0.770 0.5 0.0101 0.0192 0.0230 0.0263 0.0309 0.0325 0.0207
0.800 0.5 0.0104 0.0198 ...... 0.0271 0.0318 0.0334 0.0213
0.833 0.5 0.0107 0.0203 0.0244 0.0279 0.0327 0.0343 0.0219
0.909 0.5 0.0113 0.0214 0.0256 0.0292 0.0342 0:0359 0.0230
1.000 0.5 0.0115 0.0220 0.0263 0.0299 0.0351 0.0368 0.0236
Moments for b/a=0, 0.5, 0.8, and 1.0 from Leits.
All others, and M, at y/b = 0.25 for b/a= 1.0, from Bittner.
All values of M, from Bittner except for x/a=0.5 were obtained by interpolation from moments
for odd values of x/a, and may be in error by three or four in the last place.
Averages by Simpson's One-third Rule.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS 85
TABLE 18
SUMMARY OF MAXiMUM AND AVERAGE MOMENTS IN UNIFORMLY LOADED
SIMPLY-SUPPORTED RECTANGULAR PLATES
Poisson's Ratio=0. See Fig. 12 for notation.
b
a
0
0.250
0.333
0.500
0.667
0.714
0.770
0.800
0.833
0.909
1.000
M1,
Al,.
wub'
at
x/a =-0.5
y/b = 0.5
0.1250
0.1231
0.1172
0.0964
0.0727
0.0665
0.0595
0.0560
0.0523
0.0447
0.0368
MA.
at
y/b=0.5
0.1250
0.0651
0.0475
0.0433
0.0385
0.0362
0.0338
0.0288
0.0236
Jf.
jaf,..
at
yi/b -0.5
0
0.0174
0.0280
0.0302
0.0325
0.0334
0.0343
0.0359
0.0368
Mav
at
x/a =0.5
0
0.0111
0.0179
0.0193
0.0207
0.0213
0.0219
0.0230
0.0236
Maximum Ma-
Wb2
0.0154
0.0164
0.0185
0.0195
0.0207
0.0213
0.0219
0.0230
0.0236
at x/a=
( = 0.32)
0.16
0.30
0.36
0.50
0.50
0.50
0.50
0.50
Source
Leitz
Leitz
Leitz
Leitz
Bittner
Bittner
Bittner
Leitz
Bittner
Bittner
Leitz
(Bittner
to as the correction moments, since the actual moments in the in-
terior of a plate are equal to the moments in a uniformly loaded
simply-supported plate plus these correction moments resulting from
the edge conditions. Final moments in the interior are probably not
as accurate as those on an edge, since in some cases the simply-
supported plate moments had to be obtained from the data given
in Section 26 by interpolation. Consequently, these moments may
be in error by as much as three or four in the last place given.
Moments in the fixed span are given in Table 19 for the several
plates analyzed, and moments in the simply-supported span are
given in Table 20. It should be noted that in both tables the moments
for b/a = 0 are given for values of x/b rather than x/a, since the
length a is infinite. An exception occurs in Table 19 in the column
giving M, at x/a = 0.5. In this case the moments for b/a = 0 are for an
arbitrary location near the middle of the infinitely long span a.
Average moments are given in every case considered. In Table 19
the average moments at a fixed edge, and the average correction
moments in the interior, were obtained directly from the terms in
the series for the respective moments by means of the expression
given in Section 25. The average positive moments given in Table 19
are equal to the sum of the average positive moments that are
obtained from the data in Section 26, by interpolation if necessary,
and the average correction moments obtained as described above.
ILLINOIS ENGINEERING EXPERIMENT STATION
In Table 20, the average moments were computed from the moments
at the one-tenth points by the application of Simpson's one-third
rule. Where the moments at the one-tenth points are not given ill
this table, the average moments were obtained as the sum of the
average simply-supported plate moments and the average correc-
tion moments, which were computed by means of Simpson's one-
third rule from values at the one-tenth points.
TABLE 19
MOMENTS IN FIXED SPAN OF UNIFORMLY LOADED RECTANGULAR PLATES FIXED
ON ONE EDGE AND SIMPLY-SUPPORTED ON OTHERS
Poisson's Ratio=0. See Fig. 12 for notation. Fixed edge is at x=0 or y=0 as noted in table
headings. Average moments from terms in series.
M.
- min fixed span for
short edge at x = 0 fixed
y/b=0.5
-0.1248
-0.0644
-0.0644
-0.0034
0.0092
0.0151
0.0174
0.0181
0.0158
0.0120
-0.1216
-0.0239
0.0115
0.0217
0.0213
0.0206
0.0212
0.0236
0.0259
0.0225
-0.1185
0.0243
-0.1147
0.0280
-0.1084
0.0312
-0.0981
0.0327
-0.0916
0.0328
-0.0840
0.0146
0.0259
0.0318
0.0341
0.0329
May
-0.0832
-0.0410
-0.0161
-0.0017
0.0060
0.0098
0.0111
0.0115
0.0100
0.0076
-0.0812
-0.0145
0.0075
0.0132
0.0135
0.0130
0.0134
0.0150
0.0167
0.0169
0.0151
-0.0794
0.0153
-0.0771
0.0178
-0.0727
0.0199
-0.0664
0.0210
-0.0623
0.0210
-0.0572
0.0098
0.0168
0.0205
0.0219
0.0213
b
a
0
0.6
0.8
0.9
1.0
y
b
0
0.500
0.625*
0
0.5
0.6
0.62*
0.7
0.8
0
0.5
0
0.5
0
0.5
M- in fixed span for
wb2
long edge at y =0 fixed
x/a=0.5
-0.1250
0.0625
0.0702
-0.1210
0.0582
-0.1156
0.0539
-0.1086
0.0487
-0.1009
0.0428
-0.0922
0.0371
May
-0.1250
0.0625
0.0702
-0.0905
0.0410
0.0452
0.0453
0.0435
0.0359
-0.0836
0.0367
-0.0767
0.0321
-0.0700
0.0279
-0.0633
0.0242
Same as -hll at left
* Maximum moment obtained by algebraic interpolation using parabola fitted through three points.
b
a
0
0.500
0.555
0.625
0.714
0.833
0.909
1.000
x
a
x/b=0
0.1
0.2
0.3
0.4
0.5
0.6
0.68*
0.8
1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.85*
0.9
0
0.5
0
0.5
0
0.5
0
0.5
0
0.3
0.4
0.5
0.6*
0.7
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
f0 00
- a1 M. -
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02
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4,
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o coo
o 5oo
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ILLINOIS ENGINEERING EXPERIMENT STATION
28. Plates with Two Opposite Edges Fixed
Moments are given in this section for uniformly loaded rec-
tangular plates having two opposite edges fixed and the other two
simply-supported. These solutions are generally parallel to those
described in Section 27, and differ only in that an additional edge
is considered fixed, opposite to the single fixed edge of that section.
TABLE 21
MOMENTS IN FIXED SPAN OF UNIFORMLY LOADED RECTANGULAR PLATES FIXED
ON Two OPPOSITE EDGES AND SIMPLY-SUPPORTED ON OTHERS
Poisson's Ratio=0. See Fig. 12 for notation. All moments symmetrical about both centerlines
of plate. Long or short edges fixed as indicated in table headings.
M- in fixed span for --in fixed span for
b x short edges fixed b y long edges fixed
a a a b
y/b=0.5 Ma. x/a=0.5 M.v
0.100 0 -0.1242 ...... 0.3 0 -0.0836 ......
0.167 0 -0.1244 ...... 0.4 0 -0.0841 ......
0.200 0 -0.1250 ......
0.5 0 -0.0841 -0.0662
0.500 0 -0.1192 -0.0796 0.5 0.0416 0.0305
0.1 -0.0227 -0.0139 0.6 0 -0.0833 -0.0627
0.2 0.0128 0.0084 0.5 0.0402 0.0283
0.3 0.0224 0.0142 -
0.4 0.0237 0.0150 0.7 0 -0.0814 -0.0592
0.5 0.0234 0.0148 0.5 0.0361 0.0255
0.555 0 -0.1150 -0.0771 0.8 0 -0.0783 -0.0556
0.5 0.0272 0.0171 0.5 0.0352 0.0231
0.625 0 -0.1089 -0.0732 0.9 0 -0.0743 -0.0519
0.5 0.0300 0.0192 0.5 0.0317 0.0210
0.714 0 -0.0997 -0.0672 1.0 0 -0.0697 -0.0481
0.5 0.0320 0.0204 0.5 0.0284 0.0184
0.833 0 -0.0866 -0.0589
0.5 0.0317 0.0201
0.909 0 -0.0785 -0.0538
0.5 0.0306 0.0197
1.000 0 -0.0697 -0.0481
0.5 0.0284 0.0184
NOTE: Values of M, at various x/b for b/a=0 are same as those given in Table 19.
The method of solution, the values of b/a, and the number of terms
in the series are the same for both sets of solutions. Consequently
the discussion in Section 27 regarding the accuracy of the results
applies also herein. Average moments and positive moments were,
in general, computed by the same procedures for the two groups
of plates.
MO1MENTS IN TWO-WAY CONCRETE FLOOR SLABS
The moments are given in Tables 21 and 22, which are comparable
in content to Tables 19 and 20 respectively. In Table 21, the values
of M. for ba =0 have been omitted, as they are identical with
those given in Table 19. Also, because of symmetry, the maximum
positive moments nearly always occurred at midspan, and moments
TABLE 22
MOMENTS IN SIMPLY-SUPPORTED SPAN OF UNIFORMLY LOADED RECTANGULAR
PLATES FIXED ON Two OPPOSITE EDGES AND SIMPLY-
SUPPORTED ON OTHERS
Poisson's Ratio=0. See Fig. 12 for notation. All moments symmetrical about both centerlines
of plate. Long or short edges fixed as indicated in table headings.
-- for short edges fixed, for x/a=
b y bI
a b
0.1 0.2 0.3 0.4 0.5 M.,
0.500 0.5 0.0142 0.0395 0.0617 0.0758 0.0796 0.0463
0.555 0.5 0.0692* 0.0395
0.625 0.5 0.0562* 0.0313
0.714 0.5 0.0058 0.0185 0.0305 0.0388 0.0422 0.0230
0.833 0.5 0.0035 0.0119 0.0202 0.0260 0.0281 0.0152
1.000 0.5 0.0019 0.0064 0.0111 0.0146 0.0158 0.0084
for long edges fixed, for y/b =
0.1 0.2 0.3 0.4 0.5 M.,
0.5 0.1 0.0078
0.15 0.0071
0.2 0.0055
0.3 0.0029
0.4 9.0015
0.5 0.0005 0.0006 0.0012 0.0015 0.0016 0.0009
0.6 0.5 0.0041 0.0021
0.7 0.5 0.0071* 0.0038
0.8 0.5 0.0010 0.0035 0.0070 0.0095 0.0102 0.0052
0.9 0.5 0.0130* 0.0070
1.0 0.5 0.0019 0.0064 0.0111 0.0146 0.0158 0.0084
* Obtained by graphical interpolation from moments for other values of b/a.
NOTE: Values of MA at various x/b for b/a =0 are same as those given in Table 20.
at other values of x1 a or y/b were not computed, except for Mx in
the plate having the short sides fixed and b a =0.5.
Table 22 differs from Table 20 principally in the omission of the
moments at x a or y b between 0.5 and 1.0. Because of symmetry,
these moments need not be tabulated. The moments M. for b/a=0
are also omitted from this table, since they are identical with those
given in Table 20.
ILLINOIS ENGINEERING EXPERIMENT STATION
29. Plates with Two Adjacent Edges Fixed
Moments are given in this section for uniformly loaded rec-
tangular plates having two adjacent edges fixed and the other two
simply-supported. Moments were computed at the fixed edges and in
the interior, for plates having ratios of sides equal to 0.5 and 1.0.
The moments given herein were computed by the writers, using
a modification of the method developed by Timoshenko.1 The pro-
cedure described by Timoshenko involves the expression of moments
and rotations on the edges of the plate in terms of infinite series.
Equations are derived for the rotation at each edge in terms of the
moments on that edge and on the other edges. At a fixed edge the net
rotation must equal zero; that is, the rotation produced by the
bending moments on the various edges must be equal and opposite
to the rotation produced by the given loading on a simply-supported
plate. To satisfy this condition each term in the series for rotation
at a fixed edge is set equal to zero, and a number of equations are
obtained, one for each term in the series for each fixed edge. By con-
sidering only a finite number of terms in the series, the resulting
finite number of equations may be solved for the coefficients of the
terms in the series for edge moments.
The equations given by Timoshenko have been modified herein
so as to utilize the constants tabulated in Bulletin 304. Consider a
rectangular plate supported on all four sides and having the dimen-
sions and coordinate system indicated in Fig. 12. Let the moments
be expressed in the following terms:
Mi =E An sin -
n b
M2 = B.n sin --
n b
(11)
M3 = Cm sin --
a
mirx
M4 = E D,, sin -
m, a
wherein A., Bn, Cm, and Dm are the coefficients whose values are
to be determined. The rotations on each edge may be expressed in
a similar manner as follows:
2 S. P. Timoshenko, "Bending of Rectangular Plates with Clamped Edges," Proc. Fifth
International Congress for Applied Mechanics. Wiley, New York. 1940; pp. 40-43. See also
"Theory of Plates and Shells" by the same author (McGraw-Hill, New York, 1940), pp. 222 ff.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS 91
niry
4>) = E P- sin --
n b
D2 = Q. sin-
n b
(12)
mrx
P3 =R sin--
m a
mvx
D44 = Um sin --
m a
The rotations on each edge may. be expressed in terms of the moments
on all edges by means of the following equations relating the series
coefficients for rotations and moments.
A, Bn(k)n
Pn= - +-
N N
- (C,)n --- (C,).
a a
a 2n b2 m
+ a 2 2 E (C. - cos nrD.)
N 7r2 a2 ( b2s
( n2 + m2 -
Bn A,(k),n
Q,,= +
N N
- (C.). - (C.).
a a
a 2n b2 m(- cos mr)
+ (C,. - cos nrD.)
N rT a2 m b2
(n2 + m2 12
C. D.(k), (13)
R.= ---
N N
- (C.). -- (.).
b b
b 2m a2 n
+ N 2mb2 a2 n (A. - cos mrB,)
N rs bo' n a' \c
7 m2 + n2 -_)
b2
D,m C.(k).
UM ,= +
N N
- (C.). (C.).
b b
ILLINOIS ENGINEERING EXPERIMENT STATION
b 2m a2 n(-cos nr)
+ (A. - cos mrB.).
S(m2 + n2 V2
In the above equations the terms (C,)n and (C,)m are modified stiff-
ness factors as defined in Bulletin 304. Numerical values may be
obtained from Table 1 of that bulletin for values of (b/s) = na/b
of mb/a as indicated by the subscripts n or m, respectively. The
terms (k). and (k). are carry-over factors in the long and short
spans respectively, as defined in Bulletin 304; numerical values may
be obtained from Table 2 of that bulletin for (b/s) = na/b or mb/a,
as indicated by the subscripts n or m respectively.
The rotations of an edge of a simply-supported slab are given
by the following expressions for the coefficients of the terms in
the series: -(M')3 (1 + k.)
Pn= Qn =
N
- (C.).
a
(14)
- (MP). (1 + k.)
Rm= Urn = N
N
- (C.).
b
The term (Mf) is the fixed-edge moment as defined in Bulletin 304.
Numerical values may be obtained from the tables of that Bulletin
for various types of loading. For a uniformly distributed load of
magnitude, w, the fixed-edge moments are
4
(MPF), = - (cm)n wa2 - , for n odd only
nr
(15)
4
(MF), = - (C)m. wb2 - , for m odd only.
mr
The quantity (cm) may be obtained from Table 5 of Bulletin 304 for
(b/s) = na/b or mb/a as indicated by the subscripts n or m respectively.
It is more convenient when solving for moments in plates with
one or more fixed edges to use the relations stated above in some-
what different form. If the relations in Equations (13) and (14) are
combined to express the rotation of each edge as a function of the
loading and of the edge moments, and if these expressions are each
in turn set equal to zero, the resulting equations may be used to
express the relation between the moment on a fixed edge and the
moments on the other edges and the loading. These relations are:
MOMENTS IN TWO-WAY CONCRETE FLOOR SIABS
A,, = (M'),, (1 + k,,) - B,,k,,
2 b2 m
-n(C),, + 2
11 + mt2
(C,,, - cos nirD,,)
a-2
a-
b"2 m(- cos mr)
--- + )(C., - cos n7D,,,)
a 2 2
a-
b2 n
(n + m2
a62
a
D,, = (MF)m (1 + k,,,) - C,,,k.
2
-m(Cs),--
71
b2 n(-cos nr)
a2 (n b 2
n a2+m2 -
as 2
(A,, - cos mrB,).
The plates considered in this section have the edges at x=0 and
y = 0 fixed, and the edges at x = a and y = b simply-supported. For
these conditions, the moments M2 and M4 are equal to zero, and
thus B,=D,,=O. Equations to be solved for values of A, and C,,
may then be written using only the first and third of Equations (16).
These equations are given below in the form used in these calculations.
-An 4 a2
-- = (c,,)n (1 + k,) -
wb2 nr b2
2 b-
-n(C,)n -----
-r2 a 2
-Cm 4
= (C,,,) (1 + k,,)-
wb2 mr
2 b2
- m(C)m -- --
r2 a2
m- Cm
(n2 + 2 2 wb2
a2/
n -An
n2 + m2 6 2 wV
B,, = (MF),, (1 + k,,) - A,,k,,
2
-n(C.)n --
7 2
2
-m(C.).7-2
7r2
(16)
(An - cos mrB,)
C,. = (MF), (1 + k.,) - D,,,,,
ILLINOIS ENGINEERING EXPERIMENT STATION
In the numerical solutions, eleven terms were used for both n and m.
For b,'a = 0.5, this resulted in 22 equations involving the 22 unknown
coefficients for moment, A1, A2, A3, . . . An and C1, C2, C3, . . . Cn.
For b a = 1.0, the moments M, and Ms are equal and only eleven
equations in eleven unknowns were obtained. In each case, the equa-
tions were solved by a process of successive approximations. The
values of the coefficients thus determined are given below for the
two plates analyzed. The results are accurate to the last place given.
b/a=0.5 b/a=1.0
n or -
A. C. A.=C.
1 -0.077422 -0.125769 -0.067323
2 +0.015362 +0.011695 +0.011572
3 +0.001853 -0.007965 +0.000945
4 +0.003452 +0.006493 +0.003091
5 +0.000996 +0.000341 +0.000794
6 +0.001300 +0.003106 +0.001154
7 +0.000514 +0.000722 +0.000393
8 +0.000641 +0.001634 +0.000535
9 +0.000304 +0.000519 +0.000207
10 +0.000374 +0.000938 +0.000285
11 +0.000201 +0.000343 +0.000119
The various moments computed from the above coefficients are
given in Table 23. Negative moments on the fixed edges were ob-
tained at intervals of one-twelfth the width of the plate. Positive
moments in the interior were obtained only on certain sections, and
with one exception, only the average moment was computed. The
calculation for positive moments consisted first of determining the
moments in a simply-supported slab from the data given in Sec-
tion 26, or from the sources mentioned therein. The corrections to
these moments due to the moments acting on the edges were then
obtained by means of the procedure in Bulletin 304.
The computations described above for b a= 0.5 and 1.0 were also
made for only one term in the series expression for moment; that is,
for a single sine wave of moment on each edge. For b a =0.5 the
following edge moments were obtained:
Mi iry
at x = 0, - = -0.0743 sin --
wb2 b
at y = 0, -- = -0.1258 sin --
wb2 a
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
TABLE 23
MOMENTS IN INIFOMItLY LOADED RECTANGULAR PLATES FIXED ON Two
ADJACENT EDGES AND SIMPLY-SUPPORTED ON OTHERS
Poisson's Itatio=0. See Fig. 12 for notation. Fixed edges are at x=0 and y=0. Average moments
on edges from terms in series, all others based on calculations by Simpson's one-third rule.
b X .
I 2J 6
0.5 0
0.5
0.6
0.62*
0.7
0.8
I--
1.0 0
0.4
0.5
0.6
0.7
-12
a12
"l2
912
'8i2
i2
l12
Av
Av
Av
Av
Av
12
3i2
12
612
1i2
Av
9i2
Vi12
Av
Av
Av'
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Av
Av
M.
it62
-0.0043
-0.0216
-0.0394
-0.0571
-0.0701
-0.0787
-0.0813
-0.0782
-0.0683
-0.0524
-0.0297
-0.0487
0.0368
0.0404
0.0405
0.0390
0.0323
-0.0052
-0.0208
-0.0368
-0.0512
-0.0615
-0.0678
-0.0694
-0.0663
-0.0582
-0.0448
-0.0257
-0.0425
0.0118
0.0140
0.0030
0.0090
0.0160
0.0215
0.0249
0.0252
0.0225
0.0169
0.0089
y
0
0.5
0.6
0.7
0.8
0.85
0. 87*
0.9
0.0149
0.0147
x
a
112
?i2
912
ii2
ýi2
11i2
912
912
'912
Av
Av
Av
Av
Av
Av
Av
Av'
M.
wb'
-0.0202
-0.0582
-0.0832
-0.1038
-0.1132
-0.1180
-0.1180
-0.1140
-0.1042
-0.0853
-0.0532
-0.0813
0.0048
0.0051
0.0070
0.0094
0.0107
0.0108
0.0105
* Maximum moment obtained by algebraic interpolation using parabola fitted through three points.
ILLINOIS ENGINEERING EXPERIMENT STATION
The moment, Mx, at the middle of the short edge is thus -0.0743
from the.one-term solution, as compared with -0.0787 from the
eleven-term solution in Table 23. At the middle of the long edge the
corresponding moments, My, are -0.1258 from the one-term solution
and -0.1180 from Table 23. The accuracy of the one-term solution is
somewhat better if the average moments are considered. On the short
edge, the average moment, M,, is -0.0472 for one-term, and -0.0487
for eleven terms; while on the long edge the corresponding values of
average moment, M,, are -0.0800 and -0.0813.
For b a =1.0 the edge moment for only one term in the series
was as follows:
at x = 0, = -0.0663 sin--
wb2 b
The moment at the middle of the edge is -0.0663 for one term, as
compared to -0.0678 from Table 23, while the respective average
moments are -0.0422 and -0.0425.
It is evident from the above comparisons that the one-term solu-
tion is reasonably accurate for average moments, and since it is these
moments that are of particular interest in this Bulletin, an additional
solution was made for a plate having b/a=0.7. The edge moments
thus obtained were as follows:
Mz 7ry
at x = 0, - = -0.0729 sin -
wb2 b
May
-M- = -0.0464
wb2
My Arx
at y = 0, = -0.1010 sin -
wb2 a
May
--- = -0.0643.
wbV
30. Plates with Three Edges Fixed
Moments are given in this section for uniformly loaded rectangular
plates having three edges fixed and the other edge simply-supported.
The moments in plates having a long edge simply-supported are
based on the solutions obtained by Young' for values of b a =0.33,
0.50, 0.75, and 1.0. The moments for plates having a short edge
I Dana Young, "Analysis of Clamped Rectangular Plates," Jour. Appl. Mlech., December
1940, Vol. 7, No. 4, pp. A-139-142.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
simply-supported were obtained from solutions by the writers for
b/a= 0.50 and 0.75.
Coefficients for the terms in the series for moments on the edges
of plates having a long edge simply-supported are given by Young
in Table 3 of the paper referred to. Moments at the one-twelfth
points and average moments on the edges were computed directly
from these coefficients. The values thus obtained at the middle of
an edge differed by not more than 0.0002 from the values given by
Young. In order to compute positive moments in the interior for the
plates having b a= 0.5 and 1.0, the moments given by Young in
the form of a cosine series were converted into a sine series, and the
correction moments in the interior were obtained by means of the
TABLE 24
MOMENTS IN UNIFORMLY LOADED RECTANGULAR PLATES FIXED ON THREE
EDGES AND SIMPLY-SUPPORTED ON ONE LONG EDGE
Poisson's Ratio=0. See Fig. 12 for notation. Simply-supported edge is at y=b. See Table 25
for b/a= 1.0.
b x
a a
0.33
0.50
0.75
0
I,
b
0 12
;12
212
i i 2
I 12
412
ýi2
%,2
"'i2
Av
0.5 Av
0.6 Av
0.66* Av
0.7 Av
0.8 Av
0.9 Av
0 1'i2
I i2
*12
%i2
S i2
A12
9i2
'12
'i2
Av
uM.b2
.,b2
-0.0403
-0.0788
-0.0686
-0.0487
-0.0052
-0.0210
-0.0403
-0.0566
-0.0701
-0.0786
-0.0810
-0.0781
-0.0683
-0.0522
-0.0297
-0.0487
. 0055
0.0060
0.0063
0.0062
0.0046
-0.0067
-0.0055-
-0.0207
-0.0388
-0. 0537
-0.0658
-0.0730
-0.0749
-0.0720
-0.0630
-0.0483
-0.0276
-0.0455-
y x
b a
0 ki2
li2
912
912
Av
0 %2
3i2
M12
Av
0.3
0.6
0.64.
0.7
0.8
0
-0 0393
-0.0852
-0.1098
-0.1200
-0.1235-
-0.1244
-0.0902
-0.0207
-0.0574
-0.0856
-0.1026
-0.1117
-0.1148
-0.0726
-0.0097
-0.0327
-0.0545
-0.0705
-0.0804
-0.0838
-0.0482
* Maximum moment obtained by algebraic interpolation using parabola fitted through three points.
0.50
'i
|i
ILLINOIS ENGINEERING EXPERIMENT STATION
procedure of Bulletin 304. The simply-supported slab moments were
obtained from the sources given in Section 26.
The moments computed from Young's coefficients are given in
Table 24 for b/a =0.33, 0.50, and 0.75, and in Table 25 for b/a = 1.0.
Solutions for plates having a short side simply-supported were
obtained by the writers for b/a=0.50 and 0.75 by means of the
procedure described in Section 29. In each case there are only two
unknown moments, and terms in the series through n= 11 were
used for each. Since the moment on one edge is symmetrical and the
even values of n are thus equal to zero for that moment, only 17
equations involving 17 unknown coefficients were obtained. These
TABLE 25
MOMENTS IN UNIFORMLY LOADED RECTANGULAR PLATES FIXED ON THREE EDGES
AND SIMPLY-SUPPORTED ON ONE SHORT EDGE
Poisson's Ratio=0. See Fig. 12 for notation. Simply-supported edge is at x=a.
b x y M. y x M.
a a b wb2 b a wbl
1.0 0 42 -0.0053 0 12 -0.0056
M12 -0.0196 Hi2 -0.0193
%2 -0.0341 M2 -0.0345
fi2 -0.0454 %2 -0.0461
M% -0.0526 Mh -0.0551
M2 -0.0551 %2 -0.0601
Av -0.0307 A2 -0.0610
M2 -0.0586
%2 -0.0515
1912 -0.0399
112 -0.0231
Av -0.0381
0.4 Av 0.0080 0.5 Av 0.0137
0.5 Av 0.0090
0.6 Av 0.0095
0.7 Av 0.0097
0.8 Av 0.0092
0.75 0 M12 -0.0571 0 M2 -0.0740
Av -0.0314 %M2 -0.0750
ý12 -0.0757
Ms -0.0726
Av -0.0495
0.50 0 M12 -0.0045- 0 %M -0.0816
M2 -0.0200 Ms -0.0837
Ml -0.0343 MsI -0.0836
fM -0.0468 M2 -0.0819
M2 -0.0541 Av -0.0610
%2 -0.0571
Av -0.0314
0.1 Av -0.0004 0.5 0.1 0.0089
0.2 Av 0.0063 0.2 0.0219
0.3 Av 0.0038 0.3 0.0319
0.4 Av 0.0023 0.4 0.0381
0.5 Av 0.0013 0.5 0.0412
0.6 Av 0.0015 0.6 0.0426
0.7 Av 0.0032 0.7 0.0399
0.8 Av 0.0057 0.8 0.0321
0.85 Av 0.0071 0.9 0.0187
0.88* Av 0.0078 Av 0.0279
_0.9 Av 0.0075
* Maximum moment obtained by algebraic interpolation using parabola fitted through three points.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
equations were solved, as in the previous case, by a process of succes-
sive approximations. The terms in the series are given below for a
plate having the simply-supported edge at .r=a. If the moments
are expressed, in the notation of Fig. 12, as:
Mý nry
--at x = 0 = X.n sin--
wb2 . b
My nvx
--at y = 0 or b = Y. sin --
wb2 n a
then the coefficients X. and Y, are as follows:
b/a=0.5 b/a=0.7
n
X. Y. X, Y.
1 -0.052266 -0.092564 -0.052310 -0.077353
2 0 +0.007011 0 +0.009637
3 +0.006479 -0.009712 +0.006428 -0.002046
4 0 +0.004908 0 +0.003854
5 +0.002635 -0.000544 +0.002570 +0.000623
6 0 +0.002560 0 +0.001635
7 +0.001301 +0.000380 +0.001175 +0.000462
8 0 +0.001413 0 +0.000808
9 +0.000757 +0.000372 +0.000634 +0.000279
10 0 +0.000839 0 +0.000446
11 +0.000493 +0.000274 +0.000384 +0.000170
The moments computed from the above coefficients are given in
Table 25.
Values of the average moments on the fixed edges for other
values of b/a were needed for use in the development of the distri-
bution procedure described in Chapter III. These moments were
obtained by graphical interpolation from a curve of average moment
vs. b/a. For convenience, all of the average edge moments are sum-
marized in Table 26. The accuracy of the interpolated moments for
b/a =0.571, 0.667, and 0.800 is obviously less than that indicated
by the number of significant figures given. It is believed, however,
that these moments are not in error by more than five in the last place.
Calculation of Rotations.-Rotations of the simply-supported edge
were computed for three of the cases considered above: b/a= 1.0,
b/a=0.5 with a long edge simply-supported, and b/a=0.5 with a
short edge simply-supported. The rotation on each simply-supported
edge was expressed as a sine series, and the coefficient for each
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 26
SUMMARY OF AVERAGE MOMENTS ON FIXED EDGES OF UINIFORMLY LOADED
RECTANGULAR PLATES FIXED ON THREE EDGES
See Fig. 12 for notation. Asterisk (*) indicates moments obtained graphically from curves of
moment vs. b/a.
Average Moments Average Moments
b b
a Ma M, a Mt M
w- atX 0 -M at y = 0 - atx=0 at y = 0
iT' wb2 wb' wtrb
Long edge at y = b simply-supported Short edge at x = a simply-supported
0 -0.0487 -0.1250 0 -0.0314 -0.0833
0.333 -0.0487 -0.0902 0.500 -0.0314 -0 0610
0.500 -0.0487 -0.0726 0.571* -0.0314 -0.0580
0.571" -0.0484 -0.0654 0.667* -0.0314 -0.0532
0.667* -0.0475 -0.0557 0.750 -0.0314 -0.0495
0.750 -0.0455- -0.0482 0.800 -0.0312 -0.0470
0.800* -0.0440 -0.0440 1.000 -0.0307 -0.0381
1.000 -0.0381 -0.0307
term was computed by means of Equations (13) and (14) of Section
29. The net rotation is the sum of that due to the loading, as deter-
mined from the appropriate Equations (14), and that due to the edge
moments, from Equations (13). The series expressions are given below.
For the long edge at y = b simply-supported:
N 4's m=rx
- n= 2 1sin ---
b wb2 , a
For the short edge at x = a simply-supported:
N b, =n ry
= on sin --
a wb2 , b
The terms in the series and the average rotation for
sidered are given in the following table:
the cases con-
'Dom *n
n or m
b/a =0.5 b/a = 1.0 b/a=0.5
1 +0.017555 +0.005780 +0.003166
3 -0.000628 -0.000635- -0.000372
5 -0.000631 -0.000207 -0.000110
7 -0.000329 -0.000087 -0.000045
9 -0.000182 -0.000044 -0.000023
11 -0.000109 -0.000025 -0.000013
Av +0.0110 +0.0035 +0.0019
100
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
TABLE 27
MOIMENTS IN I'NIFORMLY LOADED RECTANGULAR PLATES FIXED
ON ALL For EDi(ES
Poisson's Ratio = 0. See Fig: 12 for notation.
, in short span, for x/a =
0.1 0.2 0.3 0.4 0.5 3i.,.
-0.0247 -0.0572 -0.0743 -0.0808 -0.0828 -0.0556
-0.0090 -0.0220 -0.0311 -0.0355 -0.0369 -0.0233
0. 0006 0.0002 -0.0007 -0.0018 -0.0020 -0. 0005
0 .0063 0.0144 0.0187 0.0212 0.0218 0.0143
0. 0094 0.0216 0.0301 0.0346 0.0357 0.0228
0. 0100 0.0237 0.0336 0.0392 0.0404 0.0255
-0.0805 -0.0518
-0.0167 -0.0434 -0.0620 -0.0723 -0.0756 -0.0464
-0.0664 -0.0389
b
0.500
0.571
0.667
0.800
1.000
-0.0247
-0.0073
0.0013
0.0054
0.0071
0.0077
-0.0389
-0.0125
0.0018
0.0089
0.0120
0.0126
-0.0482
-0.0158
0.0020
0.0111
0.0153
0.0164
-0.0513
-0.0171
0.0020
0.0118
0.0164
0.0175
-0.0290
-0.0094
0.0014
0.0067
0.0090
0.0096
b--, in long span, for y/b =
0.500 1 0
0.1
0.2
0.3
0.4
0.5
0.571 0
0.667 0
0.800 . 0
1.000 ,
0.1 0.2 0.3 0.4 0.5 MX.
1-0.0077 -0.0257 -0.0422 -0.0531 -0.0573 -0.0314
0.0010 -0.0001 -0.0017 -0.0023 -0.0024 -0.0008
0.0011 0.0041 0.0068 0.0084 0.0091 0.0050
0.0012 0.0034 0.0055 0.0071 0.0078 0.0042
0.0004 0.0019 0.0033 0.0045 0.0050 0.0025
0.0006 0.0015 0.0025 0.0036 0.0039 0.0020
-0.0571 -0.0315
-0.0079 -0.0257 -0.0422 -0.0532 -0.0570 -0.0315
-0.0559 -0.0311
Same as y1, above
The average rotation on each edge is equal to the coefficient from the
above table multiplied by wb2 - b N or wb2 a, N for the long and
short edges respectively.
31. Plates with Four Edges Fixed
Moments are given in this section for uniformly loaded rectangular
plates having all four edges fixed.
The moments given in Table 27 for b, a =0.500, 0.571, 0.667,
0.800, and 1.000 are based on the results given by Wojtaszak,'
1 I. A. Wojtaszak. "The Calculation of Maximum Deflection, Moment, and Shear for Uni-
formly Loaded Rectangular Plate with Clamped Edges," Jour. Appl. Mech., December 1937,
Vol. 4. pp. A-173-176.
-0.0077
-0.0027
0.0006
0.0019
0.0022
0.0024
b x
at a
ILLINOIS ENGINEERING EXPERIMENT STATION
modified slightly in certain cases in order to simplify the calculation
of moments in the interior of the plate. The solutions given by
Wojtaszak consist of a parabolic term plus a correction term in the
form of a cosine series. Values of the coefficients for terms in this
series are tabulated for n = 1 through 27. The expressions given by
Wojtaszak were used without change to obtain all the moments
on an edge given in Table 27, except certain values for b a =0.5
and 1.0 which are mentioned later. For the calculation of moments
in the interior of the plates having b/a =0.5 and 1.0, the expressions
for edge moments obtained from Wojtaszak were modified as follows:
First, the origin of coordinates was shifted from the middle of the
plate to a corner as in Fig. 12, and the cosine series was converted
to a sine series referred to the new axes. The parabolic term was
then expanded into a sine series which was added term by term to
the other. The result was an expression for edge moments involving
only a sine series whose terms were easily computed from Wojtaszak's
tabulated coefficients and the expansion of the parabolic term. Cor-
rection moments in the interior of the plate were then computed by
means of the procedure in Bulletin 304. Positive moments in the
simply-supported plate at the one-tenth points were obtained from
Leitz. (See Section 26.) Moments at the one-tenth points on the edges
for b/a= 0.5 and 1.0 were computed from these modified expressions.
Average moments on an edge were obtained by the integration of
Wojtaszak's equations for moment. Average moments on sections in
the interior were obtained by the application of Simpson's one-third
rule to the values at the one-tenth points.
Certain of the moments given in Table 27, especially those at
the middle of an edge or at the center of the plate, have also been
computed by other investigators. Wojtaszak gives moments for the
middle of the long edge which differ from those given herein by not
more than 0.0001. Young1 has computed moments for b/a =0.500,
0.667, and 1.000, based on Wojtaszak's solutions. His values for
moments at the middle of an edge are not more than 0.0001 different
from those in Table 27. The difference for moments at the center
of the plate is usually not more than 0.0001, but for b/a =0.5 is
as high as 0.0004. Some errors are introduced, however, in making
the comparisons since the values given by Young are for Poisson's
ratio equal to 0.3, and are expressed in terms of the square of the
long side instead of the short side as used herein. In general the
agreement is satisfactory, but where differences exist, the interior
I Dana Young, "Analyses of Clamped Rectangular Plates," Jour. Appl. .Iech., December
1940, Vol. 7, No. 4, pp. A-139-142.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
moments given by Young are probably more nearly correct since
they were obtained by a more direct calculation.
Moments at the one-tenth points on the edges and in the interior
of a square plate are given by Leitz.1 The values given, however,
differ from those in Table 27 by as much as 0.0009. At the center
of the plate the coefficient given by Leitz is 0.0184, as compared
to 0.0175 herein and 0.0176 from Young's results. In view of this
comparison it is believed that the values given in Table 27 are more
nearly correct than those obtained by Leitz. However, the solution
by Leitz is more complete, and in some ways more useful, if a lower
degree of accuracy is acceptable. He has tabulated, in addition to
M, and M,, the twisting moments, My, and the principal moments
and their directions, all for Poisson's ratio equal to zero.
The results given by Hencky2 for b/a = 1.0 are in good agreement
with those in Table 27, and with those computed by Wojtaszak and
Young. His values for b/a =0.5, however, differ considerably and
are believed to be in error. The reason for this is that only the terms
through n = 11 were used in his calculations for b/a = 0.5. For
b/a = 1.0, the terms through n =25 were used, and good agreement
would be expected with Wojtaszak's solution for terms through n = 27.
Maximum and average moments on the edges are summarized
in Table 28 for a number of values of b /a. The additional moments
given were taken from the results obtained by Evans,3 and are
believed to be as accurate as the values from Table 27. Evans com-
puted the terms through n = 11, and obtained a number of additional
terms by extrapolation based on the observed trends. He states that
the values are not in error by as much as 0.1 percent, which corre-
sponds to less than 0.0001 for the moments given.
' N. Leitz, "Berechnung der eingespannten rechteckigen Platte," Zeits. fuir Math. u. Phys.,
Bd. 64, 1916.
2 H. Hencky, "Der Spannungszustand in rechteckigen Platten." Oldenbourg, Munich, 1913.
'T. H. Evans, "Tables of Moments and Deflections for a Rectangular Plate Fixed on All
Edges and Carrying a Uniformly Distributed Load," Jour. Appl. Mech., March 1939, Vol. 6.
No. 1, pp. A-7-11.
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 28
SUMMARY OF MAXIMUM AND AVERAGE MOMENTS ON EDGES OF UNIFORMLY
LOADED RECTANGULAR PLATES FIXED ON ALL EDGES
Coordinate system and notation is that of Fig. 12.
wb-, on long edge -b2, on short edge
b Basis for
a Moments
At Middle Average At Middle Average
of Edge Moment of Edge Moment
0 -0.0833 -0.0833
0.500 -0.0828 -0.0556 -0.0573 -0.0314 Wojtaszak*
0.526 -0.0822 -0.0571 Evans
0.556 -0.0812 -0.0571 Evans
0.571 -0.0805 -0.0518 -0.0571 -0.0315 Wojtaszak*
0.588 -0.0799 -0.0571 Evans
0.625 -0.0780 -0.0571 Evans
0.667 -0.0756 -0.0464 -0.0570 -0.0315 Wojtassak*
0.714 -0.0726 -0.0568 Evans
0.750 -0.0417 -0.0314 Interpolatedt
0.769 -0.0687 -0.0563 Evans
0.800 -0.0664 -0.0389 -0.0559 -0.0311 Wojtaszak*
0.833 -0.0639 -0.0554 Evans
0.909 -0.0581 -0.0538 Evans
1.000 -0.0513 -0.0290 -0.0513 -0.0290 Wojtaszak*
* Moments given are those computed by authors from Wojtaszak's solution. In some cases
they differ by 0.0001 from values given by Wojtaszak.
t Obtained by graphical interpolation from plot of moment vs. b/a.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
APPENDIX B
MOMENTS IN UNIFORMLY LOADED CONTINUOUS
RECTANGULAR PLATES
32. Introduction
Solutions are given in this Appendix for moments in uniformly
loaded rectangular plates continuous in two directions over rigid
supporting beams. The results set forth in the following sections
were used extensively in the derivation of the constants for the dis-
tribution procedure and in the verification of the procedure as a whole.
Moments are given for three continuous slabs, designated as
Slabs I, II, and III, plans of which are given in Fig. 16. In each
case the structure has been assumed to carry a uniform load, w,
distributed over the entire area of all panels. The beams are assumed
to be nondeflecting but to have no torsional stiffness. The exterior
edges of the slabs are thus simply-supported.
The solutions for edge moments for Slabs I and II were obtained
by C. W. Pan,' and the writers are indebted to Professor L. C. Maugh
of the Civil Engineering Department of the University of Michigan
for his kindness in permitting their presentation and use herein.
The calculations for Slab III were made by the writers, using the
equations and procedure described in Section 29 of Appendix A.
In all slabs, the positive moments in the interior of the panels were
computed by means of the procedure described in Bulletin 304.
Moments due to the uniform load on a simply-supported slab were
obtained from the sources mentioned in Section 26 of Appendix A.
The notation used throughout this Appendix is generally the
same as that of Figs. 12 and 16. Panels in the various slabs are
identified by the letter designations indicated on Fig. 16, while
points within the panels are referred to the coordinate system of
Fig. 12. The origin of coordinates is always taken at the lower left-
hand corner of the panel, with the x-axis extending horizontally and
the y-axis vertically. Irrespective of their relative lengths, the span
lengths are designated as b in the y-direction and a in the x-direction,
and points in the panel will therefore always be located by the
coordinates x/a and y/b.
All the numerical results presented in the following sections are
for a value of Poisson's ratio equal to zero.
' C. W. Pan. "Analysis of Continuous Slabs," a dissertation submitted in partial fulfilment
of the requirements for the degree of Doctor of Science in the University of Michigan, Ann
Arbor, Michigan, 1939.
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 29
MOMENTS IN UNIFORMLY LOADED CONTINUOUS SLAB HAVING NINE
SQUARE PANELS: SLAB I
Poisson's Ratio=0. For notation see Figs. 12 and 16. Values in table are average moments in
pounds for uniform load w= 100 lb per sq ft and dimensions of panels given in Fig. 16.
Panel x y M. L y x. M
a b b a
ABCD 0.2 Av +145 0.2 Av +145
0.3 Av +153 0.3 Av +153
0.4 Av +150 0.4 Av +150
0.5 Av +134 0.5 Av +134
1.0 Av -409 1.0 Av -409
0.5 -658 0.5 -658
BBDD 0 0.5 -658 0.1 Av + 66
Av -409 0.2 Av + 86
0.3 Av + 89
0.4 Av + 87
0.5 Av +143 0.5 Av + 81
1.0 Av -298
0.5 -573
DDDD 0 0.5 -573 0 0.5 -573
Av -298 Av -298
0.5 Av +104 0.5 Av +104
33. Analysis of Slab with Nine Square Panels
Moments are given in this section for a uniformly loaded con-
tinuous slab having nine square panels arranged in three rows of
three panels each. The beams are assumed to be nondeflecting, and
to have zero torsional stiffness. The notation used and the dimensions
of the slab are those of Slab I in Fig. 16. A uniform load w of 100
lb per sq ft over the entire area of the slab is considered. The struc-
ture analyzed .is symmetrical about both centerlines and moments
are therefore given for three panels only.
The moments given in Table 29 are based on the solutions ob-
tained by C. W. Pan and reported in the thesis referred to in the
preceding section. The method of analysis used has been described
in a paper by L. C. Maugh and C. W. Pan.' Expressions in terms of
infinite series are written for the slopes on all edges of each panel,
in terms of the restraining moments on all edges and the applied
loading. At each support, the expressions for edge slopes in the two
adjacent panels are equated term by term; the resulting equations
are solved for the desired restraining moments, which are also ex-
pressed as a trigonometric series. In Pan's solution, terms in the
series for moment through n=3 were obtained as described above,
and additional terms through n= 5 were computed by an approxi-
mate procedure.
' L. C. Maugh and C. W. Pan, "Moments in Continuous Rectangular Slabs on Rigid
Supports," Trans. ASCE, Vol. 107 (1942), p. 1118.
106
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
All moments given in Table 29 for the edge of a panel were ob-
tained directly from the five terms in the series given by Pan.
Average moments on sections in the interior were computed from
the edge moments and the moments in a uniformly loaded simply-
supported slab. In this latter calculation, average moments in the
simply-supported slab were obtained by means of Simpson's one-
third rule from the solutions by Leitz, referred to in Section 26.
Correction moments in the interior were then obtained from the
terms in the series for edge moments using the procedure described
in Bulletin 304, and average values were obtained either by Simpson's
one-third rule or by direct summation, depending on the nature of
the expression.
34. Analysis of Slab with Fifteen Unequal Panels
Moments are given in this section for a uniformly loaded con-
tinuous slab consisting of fifteen unequal panels having the dimensions
given in Fig. 16 for Slab II. The supporting beams are assumed to
be nondeflecting, and to have zero torsional stiffness. The uniform
load w on the entire area of the slab is taken as 100 lb per sq ft.
The moments given in Table 30 are based on the solutions ob-
tained by Pan, and were computed from the terms in the series for
edge moments in exactly the same way as those in the preceding
section. The analysis of this slab is similar in every respect to that
for the slab having nine square panels as described in Section 33.
The analysis of the slab considered in this section is described
in the ASCE paper by Maugh and Pan, referred to in the preceding
section, and values are given therein for moments at the middle
of each edge and at the middle of each panel. The edge moments
agree with those in Table 30; no moments are given herein corre-
sponding to those at the centers of the panels.
35. Analysis of Slab with Nine Unequal Panels
Moments are given in this section for a uniformly loaded con-
tinuous slab consisting of nine unequal panels having the dimensions
given in Fig. 16 for Slab III. The supporting beams are assumed to
be nondeflecting and to have zero torsional stiffness. The uniform
load w over the entire area of the slab is taken as 100 lb per sq ft.
The analysis of this slab was made by means of a procedure
similar to that used by Pan and described briefly in Section 33. Slopes
on the continuous edges of each panel were expressed in terms of the
restraining moments and the applied loading by means of Equations
(13) and (14) of Section 30. Since both the slab and loading are sym-
metrical, only four equations for edge slopes were needed, as follows:
For slope at edge BD of panel ABCD,
For slope at edge BD of panel BBDD,
For slope at edge DD of panel BBDD,
For slope at edge DD of panel DDDD.
TABLE 30
MOMENTS IN 'NIFORMLY LOADED CONTINroUS SLAB HAVIN F'IFTE.N
UNEQUAL PANELS: SLAB II
Poisson's Ratio=0. For notation see Figs. 12 and 16. Values in table are moment. in pounds
for uniform load t = 100 lb per sq ft and dimension- of panels given in Fig. 16.
Panel
aL b
a b
ABDE 0.1 Av + 321
0.2 Av + 517
0.3 Av + 606
0.38* Av + 636
0.4 Av + 635
0.5 Av + 579
1. 00.5 -2056i
Av -1370
BCEF 0 0.5 -2056
Av -1370
0.4 Av + 416
0.5 Av + 481
0.54* Av + 487
0.6 Av + 474
S 0.7 Av + 361
1.0 0.5 -1214
Av - 850
CC'FF' 0 0.5 -1214
DEDE
Av - 850
0.5 Av + 207
0.1 Av + 2119
0.2 Av + 278
0.27* Av + 282
0.3 Av + 281
0.4 Av + 267
0.5 Av 4- 256
1.0 0.5 -1217
Av - 785
0.1 Av + 166
0.2* Av + 185
0.3 Av + 166
0.4 Av + 144
0.5 Av + 138
1.0 0.5 -1005
Av - 587
0.1* Av + 64
0.15 Av + 47
0.2 Av + 18
0.3 Av - 25
0.4 Av - 42
0.5 Av - 38
1.0 112 - 3
212 - 78
11 1 - 215
'2 -- 351
`s 1 - 409
1Av - 426
Av - 209
0. 1 Av + 41 0 0.5 -1217
0.2 Av + 12 Av - 785
0.3 Av - 22
0.4 Av - 36 0.5 Av + 148
0.5 Av - 34
1.0 !i2 + 18
12a - 58
9i2 - 208
,i2 - 336
'i2 - 388
912 - 395
Av - 194
EFEF 0 Same as DEDE 0 0.5 -1005
at x/a=1.0 Av - 587
0.1 Av + 5
0.2 Av + 61 0.5 Av + 160
0.23* Av + 64
0.3 Av + 45
0.4 Av + 20
0.5 Av + 7
1.0 0.5 - 540
Av - 306
FF'FF' 0 0.5 - 540 0 Same as CC'FF'
Av - 306 at y/h= 1.0
0.5 Av + 133 0.5 Av + 110
* Maximum moment obtained by algebraic interpolation using parabola fit ted through three points.
MOMENTS IN TVWO-WAY CONCRETE FLOOR SLABS
The expressions for edge slopes in the two adjacent panels at
edges BD and DD were then equated term by term, and two sets
of equations were obtained, involving the two sets of coefficients
for terms in the series for moments on those edges. In the numerical
solution, terms in the series through n =ll were used for both
moments. Since the even-numbered terms in the series for moment
on edge DD are equal to zero because of symmetry, only 17 equa-
tions involving 17 unknown coefficients were obtained. These were
solved by a process of successive approximations identical with that
used for the solutions described in Sections 29 and 30 of Appendix A.
Four cycles of substitution (requiring four and one-half hours' work
with a computing machine) were required to obtain terms in the
series accurate to the second decimal place, corresponding to three
to six significant figures. Because of the limited number of terms
considered, the edge moments in Table 31, other than the average
TABLE 31
MOMENTS IN I'NIFORMLY LOADED CONTINUOUS SLAB HAVING NINE
UNEQUAL PANELS: SLAB III
Poisson's Ratio=0. For notation see Figs. 12 and 16. Values in table are moments in pounds
for uniform load tr= 100 lb per sq ft and dimensions of panels given in Fig. 16.
Panel X -1 - ,
ABCI) 0.2 Av + 591 Same as .1, because
0.3 Av + 683 of symmetry about
0.4* Av + 707 diagonal
0.5 Av + 681
1.0 '2i - 718
2 iL -1209
3i2 -1538
,1a -1732
'12 -1818
'i2 -1813
712 -1710
12, -1521
9i2 -1216
19i2 - 821
l!i2 - 326
Av -1209
BBDD 0 Same as ABCD 0. 1 Av + 25
at xr/a 1.0 0.15 Av - 1
0.2 Av - 37
0.5* Av - 35 0.3 Av - 94
0.4 Av -123
0.5 Av -122
1.0 I,2 + 26
,1i - 55
12 -146
1ts -232
5j2 -283
912 -304
Av -139
DDDD Same as if, because 0 Same as BBDD
of symmetry about at y/hb= 1.0
diagonal 0.5* Av +164
* Maximum positive moment.
110 ILLINOIS ENGINEERING EXPERIMENT STATION
moments, are not equally accurate, and may be in error by as much
as 2 or 3 in the last place. The average moments on an edge are
believed to be correct to the number of places given. However,
because of the several corrections involved and because of uncer-
tainties regarding the accuracy of average moments for the simply-
supported slab, the average moments given for points in the interior
of the panels may be in error by as much as 5 in the last place.
Average moments and moments at the one-twelfth points on an
edge were computed directly from the terms in the series. Moments
in the interior of a panel were obtained from the edge moments and
the simply-supported slab moments, using the procedure of Bulletin
304 in the manner described in Section 33.
A second analysis of this slab was made using only the terms in
the series through n=3. Such an analysis is relatively simple to
carry out, and it was desired to compare the results obtained with
those from the more refined calculations using eleven terms. Moments
at the one-twelfth points of edges BD and DD obtained from the
two solutions are compared in Table 32. The agreement is seen to be
excellent for average moments; and, in general, the three-term solu-
tion yielded results which would be satisfactory for many purposes.
With the exception of the one-twelfth and eleven-twelfths points, the
maximum difference for edge BD is about 4 percent plus or minus.
Larger differences, however, were obtained for edge DD.
TABLE 32
COMPARISON OF EDGE MOMENTS IN SLAB III COMPUTED FOR ELEVEN TERMS
AND FOR THREE TERMS IN SERIES
Poisson's Ratio = 0. See notes in Table 31. Moments for eleven terms are from Table 31.
M. on edge BD M, on edge DD
Il x
I a
11 Terms 3 Terms 11 Terms 3 Terms
M2 - 718 - 624 Mi2 + 26 - 3
j2 -1209 -1163 42 - 55 - 37
M2 -1538 -1557 M2 -146 -113
Mi -1732 -1785 l$2 -232 -212
M2 -1818 -1860 M2 -283 -296
2a -1813 -1813 %2 -304 -329
A2 -1710 -1676
42 -1521 -1468
%2 -1216 -1191
19i2 - 821 - 846
1%2 - 326 - 441
Av -1209 -1210 Av -139 -138
10.oMlNTSF IN TW\O-WVAY CONCIETE '1.0011O Sl.ABS
APPENDIX C
MOMENTS IN RECTANGULAR PLATES WITH
CONCENTRATED LOADS
36. Introduction
A limited number of solutions for moments in rectangular plates
subjected to concentrated loads are given in this Appendix. This
type of loading is not considered in the main body of this bulletin,
and is not usually considered in the design of two-way concrete
floor slabs for buildings. Nevertheless it was deemed worthwhile to
investigate, at least partially, tile applicability of the moment distri-
bution procedure to plates carrying concentrated loads.
TABLE 33
EDGE MOMENTS FOR CONCENTRATED LOAD AT CENTER OF RECTANGULAR PLATES
HAVING ALL EDGES FIXED
Moments from solution by Young (Jour. Appl. Mech., September, 1939). See Fig. 12 for coor-
dinate system and notation. Values in table are moments in terms of concentrated load P and length
of short span b.
M, at y=0 or b M. at x=0 or a
M 31"
a tit Av Total -P Av Total -
x/a=0.5 y/b =0.
0 -0. 1677* 0 -0.1250* 0 0 Same as
0.500 -0.1674 -0.0611 -0.1222 -0.0162 -0.0066 Average
0.555 -0. 16li7 _-0.0656 -0.1181 -0.0263 -0.0116
0.625 -0.1651 -0.0695 -0.1112 -0.0425 -0.0192
0.715 -0.1604 -0.0713 -0.0998 -0.0648 -0.0299
0.833 -0.1490 1 -0.0688 -0.0826 -0.0935 -0.0438
1.000 -0,1257 1 -0.0590 -0.0590 -0.1257 -0.0590
* Computed from Bulletin 304.
Moments were computed in rectangular plates having three or
four edges fixed, and loaded with a single concentrated load at
various locations. Two values of the ratio of sides, b/a, were con-
sidered: 0.5 and 1.0. Three load positions in the central portion of
the plate were considered for the square plates (b a= 1.0). Analyses
were made for a load at each of these positions for the condition of
four edges fixed, but for a load at only two of them for the case of
three edges fixed. Only a single load at the middle of the plate was
considered for tile rectangular plates having b a = 0.5 and either four
edges fixed or three edges fixed with the long edge simply-supported.
The moments given in Sections 38 and 39 were obtained by
the writers by means of the procedure described in Section 29,
Appendix A. Solutions obtained by other investigators, some of
ILLINOIS ENGINEERING EXPERIMENT STATION
which are for the same cases as those considered herein, are dis-
cussed in Section 37. The last section contains a brief discussion of
the applicability of the distribution procedure described in Chapter II
to slabs with concentrated loads.
37. Previous Work
The most important previous solutions for concentrated loads in
fixed rectangular plates are those reported by Young' for a central
load on plates fixed on all edges. The method of calculation was that
proposed by S. Timoshenko and referred to in Section 29, Appen-
dix A. The results are given as coefficients for the terms in the series
for edge moments for plates having values of ba =0.500, 0.555,
0.625, 0.715, 0.833, and 1.000. For b/a=0.500 and 1.000, terms in
the series through n = 13 are given; for the other values of b/a only
the terms through n = 9 were obtained.
Maximum moments at the middle of the edges, average moments,
and total moments are given in Table 33. The moment My at the
middle of the long edge was given by Young; the other moments
were computed by the writers from the terms in the series.
Moments in the plate with b/a = 1.0 have also been obtained by
Hencky,2 using terms in the series through n = 11. His results are in
reasonably good agreement with those obtained by Young.
Analyses of plates having b/a = 0.5 and 1.0 were also made by
the writers, using only the terms through n= 5. The moments ob-
tained differ somewhat from those obtained by Young for a greater
number of terms. The two sets of values may be compared by
reference to Tables 35 and 38.
Other solutions for concentrated loads on plates with fixed edges
have been reported by Young in another paper.3 Two of the cases
considered in that paper are of interest here. In the first, moments
are given at the middle of an edge for the following conditions:
b/a=0.5, three edges fixed with a long edge simply-supported, and
a concentrated load at the middle of the plate. Terms in the series
are not given, however, and it was necessary to repeat this solution
in Section 39 in order to obtain data for the calculation of average
moments and moments at other points on the edges. The values
obtained by Young and by the writers may be compared by reference
to Table 38.
The other solution of interest here is for a square plate having
I Dana Young, "Clamped Rectangular Plates with a Central Concentrated Load," Jour.
Appl. Mech., September 1939, Vol. 6, No. 3, pp. A-114-116.
2 H. Hencky, "Der Spannungszustand in rechteckigen Platten." Oldenbourg, Munich, 1913.
3 Dana Young, "Analysis of Clamped Rectangular Plates," Jour. Appl. Mech., December 1940,
Vol. 7, No. 14. pp. A-139-142.
112 .
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
b/a = I.0 - P/ates with Four Edqes FAied
b/a' /.0-P/ates with Three Edges F'xed
FIG. 21. EDGE CONDITIONS AND LOCATIONS OF LOADS
FOR PLATES ANALYZED IN APPENDIX C
four edges fixed and loaded at the quarter-point of one centerline.
Again, terms in the series are not given but moments are tabulated
for points at the middle and quarter-points of the edges equidistant
from the load, and at the middle of the other edges. These moments
are not presented herein.
38. Moments in Square Plates
Moments are given in this section for a concentrated load at
various locations on square plates supported on all sides and having
either three or four edges fixed. Moments were computed only at
the edges of the plates, but terms in the sine series for edge moment
are given so that moments in the interior may be obtained, if desired,
by means of the procedure described in Bulletin 304. Five plates were
analyzed, having the various combinations of edge conditions and
locations of load indicated in Fig. 21. Loads at positions (a) and
ILLINOIS ENGINEERING EXPERIMENT STATION
(c) were considered for both three and four edges fixed, while a
load at position (b) was considered only for the condition of four
edges fixed.
All results given herein were obtained by the procedure described
in Section 29 of Appendix A, using Equations (16). For the case of
a concentrated load P on the plate at a point .r = i, y = v, the quantity
(MF3) in Equations (16) is given by the following expressions:
a nry
(MF),, = - 2P- sin -- (C.v),,
b b
b m7ru
(MF) = - 2P- sin (C),,,
a a
where (Cm3) is obtained from Table 3 of Bulletin 304 for (b/s) in the
table equal to na/b or mb/a as indicated by the subscripts n and m
respectively, and (v b) in the table equal to v/b, 1-v/b, u /a, or
1- u/a, depending on the edge being considered. It should be noted
that the term (MF) (1+k) in Equation (16) must be separated into
its component parts, (MF) and (kMF), if the load is not symmetrical,
since the value of (MF) for one edge is obtained from Bulletin 304
for (v/b)=u/a or v/b, and for the opposite edge for (v/b)= 1 -u a
or 1-v/b. That is, the fixed-edge moments on two opposite edges
will not be equal except for symmetrically placed loads.
Only the terms in the series through n = 5 were considered, and
from 3 to 15 equations were obtained for the various plates analyzed.
These were solved, as before, by a process of successive approxima-
tions, continued until values of the coefficients were obtained correct
to the fifth decimal place.' Terms in the series are given in Table 34.
Average moments and moments at the twelfth points on the
edges, computed from the terms in the series, are given in Table 35
for four edges fixed and in Table 36 for three edges fixed. Although
these moment coefficients are given to four decimal places, they are
not that accurate because of the small number of terms in the series
used. Some idea of the probable error may be obtained by comparing
the values for a central load computed herein using 5 terms with
those computed by Young using 13 terms, as given in parentheses
in Table 35. The difference is only about 1 percent for the maximum
moment and considerably less for the average moment. However, for
unsymmetrical cases the values given herein may be more in error.
1 The time required for solution of the equations, using a fully automatic electric computing
machine, varied from 35 to 213 min, depending on the number of equations. The time per
equation varied from 10 to 20 min.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS 115
TABLE 34
TERMS IN SERIES FOR EDGE MOMENTS FOR SQUARE PLATES
WITH CONCENTRATED LOADS
See Fig. 21 for notation. Values in table are coefficients of sin nrx/a or sin nry/b.
Coefficients for Edge Moment M/P
Edge
Conditions
Four
Edges
Fixed
Three
Edges
Fixed
M, at y = 0
Same
-0.1690
0
+0.0526
0
+0.0003
Same as
MA at
x=0
-0.1090
0
+0.0284
0
+0.0048
-0.1244
-0.0578
-0.0114
+0.0181
+0.0135
Mfr at y = b
Same
-0.0324
0
+0.0189
0
+0.0056
Same as
M, at
X=a
0
0
0
0
0
0
0
0
0
0
39. Moments in Rectangular Plates with b/a = 0.5
Moments are given in this section for a concentrated load at
the center of a rectangular plate having b/a=0.5 and supported on
all four sides. Two cases are considered: a plate having all four
edges fixed, and one having three edges fixed and the long edge at
y = b simply-supported. All calculations for moments in this section
were made by exactly the same procedure as that described in Sec-
tion 38 for the square plates. The same equations from Section 29
were used, terms in the series through n=5 were considered, and
the accuracy of the results is the same as that discussed in the
preceding section.'
Terms in the sine series for edge moments are given in Table 37.
No moments were computed at points in the interior of the plate,
but the coefficients in this table may be used with the method of
Bulletin 304 to compute correction moments in the interior if desired.
1 In the solution for the plate with three edges fixed, eight equations were obtained and
the time required for their solution by successive approximations was 52 min.
Location of
load n
-i
-Mat. =0 M, at x=a
-0.1025 Same
Same as
M. at
x=0
-0.0277
-0.0039
+0.0068
+0.0013
+0.0014
Same as
M. at
x=0
-0.0341
+0.0007
+0.0050-
+0.0023
+0.0009
(a)
(b)
(e)
(a)
(b)
1
2
3
4
5
1
2
3
4
3
5
1
2
3
4
5
1
2
3
4
5
0
+0.0264
0
+0.0043
-0.0742
-0.0156
+0.0124
+0.0076
+0.0036
-0.1227
-0.0598
+0.0109
+0.0181
+0.0134
-0.1279
+0.0191
+0.0182
+0.0046
+0.0021
-0.1300
-0.0541
+0.0082
+0.0196
+0.0126
ILLINOIS ENGINEERING EXPERIMENT STATION
Average moments and moments at the twelfth points on the
edges are given in Table 38. The probable accuracy of these moments
may be estimated by comparing them with the more precise values
obtained by Young using 13 terms in the series. Young's values are
given in parentheses in the table.
40. Applicability of Distribution Procedure to Plates
with Concentrated Loads
Although the data given in the preceding sections are too limited
to support any definite conclusions, certain of the moments tabulated
may be used to make a rough check on the applicability of the carry-
TABLE 35
EDGE MOMENTS FOR CONCENTRATED LOAD AT VARIOUS LOCATIONS
ON SQUARE PLATE WITH FOUR EDGES FIXED
See Fig. 21 for notation. Values in parentheses are moments from solution by Young (see Se,-
tion 37).
Edge Moment ---
PF
x_ Mi
a P
Same as M,
ýi2
5i2
512
'52
M12
A12
Av
1.0 142
112
912
912
ý12
942
1912
1142
Avi
A,
name as 31~ at 11/52.U 0 I Same as if. at I/a = 0
Edge Moment
Location of
Load
(a)
(b)
x
a
0
or
1.0
0
or
1.0
0
or
1.0
0
12
b
HM2
ý12
A42
is2
%i2
Av
"2
Mi2
l12
M2
M2
M%
1942
142
Av
Af.
P
-0.0037
-0.0227
-0.0569
-0.0925
-0.1166
-0.1246
(-0.1257)
-0.0591
(-0.0590)
-0.0081
-0.0297
-0.0619
-0.0876
-0.0940
-0.0830
-0.0650-
-0.0471
-0.0307
-0.0161
-0.0059
-0.0441
-0.0254
-0.0799
-0.1484
-0.1854
-0.1683
-0.1201
-0.0772
-0.0504
-0.0287
-0.0076
+0.0032
-0.0741
-0.0063
-0.0318
-0.0825
-0.1466
-0.2000
-0.2213
-0.0964
+0.0105-
+0.0056
-0.0135
-0.0330
-0.0432
-0.0457
-0.0159
-0.0019
-0.0086
-0.0196
-0.0297
-0.0343
-0.0332
-0.0281
-0.0206
-0.0118
-0.0041
-0.0002
-0.0160
-------
-1
Same as M, at y/b - 1.0
1
Same as M. at x/a = 0
0 1
TABLE 3(6
IG.: .1OMENTS. FOR C(ONIENTRATEI) LOAD AT VARHIOI' LOCATIONS ON SQUARE
PLATE WITH THREE EDGEs FIIXEI)
Se,, Fig. 21 for notation. Edge at t= b is s inply-suppor 'd.
IEdge 1Monent .1dge .Moment -_
o,.ad
.r .,_ +1t, ,l x .V ,
I I , .' - If7
S i -.11045 0 L12 -0.0035
or -' . -0.11241 21 -0.0237
1 0 ' -( 0,.05)!1 -0.0604
12 -0. 11100 4 2 --0.0985
' -0. 303 ' -0.1241
S -0. 1140 0 i : -0.1326
12 -0. 1414 Av -0.0628
2 -) -0.1251
! -0.0982
"' 1_ -l-0.10652
St2- - )0.0318
Av -0. 0773
) I 1 -0.0257 (I li -0.0253
l -11.08114 212 F -0.0802
11_ -0). 1492 3'12 -0.1494
1 I -0.1874 41 -0.1870
'12 -. 1722 "' -0.1703
12 --0. 12l56 612 -0.1223
; -0.0841 712 -0.0792
- 0. 0597 8 2 -0.0520
4).2 -0.0410 1i2 -0.0297
'ul -0.0207 1 i2 -0.0079
S. - 0.0051 1 2 +0.0032
Av -40 0794 Av -0.0751
1 .0 . -0.0021
12 -0.0090
3' a 2 -0.0205
A2, -0.0317
'( 2 - -0. 0379
2 -0.0382
;12 --0.0346
12 -0.0289
9 1 2 --0. 0219
1142 -0.0142
S1 2 4 -0.0067
Av -0.0205
TABLE 37
TEiRMS IN SERIES FO EI)GE MOMENTS FOR RECTANGULAR PLATES
WITH CONCENTRATED LOAD: b/a =0.5
See Fig. 21 for notation. Values in table are coefficients of sin nz(x/a or sin nry/b. Concentrated
load I' at celter of plate.
i( C(oefficients for ldge Moment .i/P
Condi- i Loatimi
tiol- *". " --a- --
s .l ,at X=0 .iM, at x=a11 f, at /y=0 .1V, at y=b
Four (a) 1 -0.011Hi Same as -0.1101 Same as
EdIges 2 0 31 at 0 11, at
Fixed 3 + 0.0045 .r=0 +0.0460 1 =0
4 0 0
5 +0 0010 -0.0085
Three (a) 1 -0.0344 Samne as -0.1477 0
Edge' 2 +0.0112 .1, at 0 0
Fixed 3 +0.003i .r = 0 +0.0536 0
4 +0.0015- 0 0
5 +0.0008 -0.0068 0
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 38
EDGE MIoMENTS FOR CONCENTRATED LOAD AT CENTER OF RECTANGULAR PLATE
WITH THREE OR FOUR EDGES FIXED: b/a=0.5
See Fig. 21 for notation. Edge at y= b is simply-supported for plate with three edges fixed. Values
in parentheses are moments from solution by Young (see Section 37).
Edge Moment .' .
P
Edge Moment ---
P
tions x y 1 . y x M
a b P b a P
Four 0 ti2 +0.0012 0 4i2 -0.0043
Edges or I i2 -0.0008 or * i2 -0.0134
Fixed 1.0 h12 -0.0057 1.0 | 2 -0.0393
1i2 -0.0109 12 -0.0879
12 -0.0142 412 -0.1410
%i2 -0.0152 Ii2 -0.1646
(-0.0162) (-0.1674)
Av -0.0063 Av -0.0614
(-0.0066) (-0.0611)
Three 0 t'2 +0.0013 0 112 -0.0069
Edges or 12 -0.0022 z2 -0.0237
Fixed i 1.0 42 -0.0112 12 I -0.0617
'i2 -0.0221 ti2 -0.1220
12 -0.0313 12 -0.1823
12 -0.0372 i2 -0.2081
(- 0.0389) (-0.2115)
F12 -0.0399 Av -0.0835
»i2 -0.0389
9i2 -0.0335-
. 19i2 -0.0241
fi2 -0.0124
Av -0.0210
over factors from Section 6 to the edge moments resulting from
concentrated loads located in the middle third of the plate. More-
over, the coefficients given in Tables 35 and 38 may be used for
the calculation of average fixed-edge moments in plates loaded with
concentrated loads. No data are available for use in checking the
applicability of the stiffness factors proposed for uniformly loaded
plates, but is probable that they will be satisfactory if the carry-over
factors are.
To check the carry-over factors it is necessary that average
moments on all edges be known for plates with four edges fixed and
with three edges fixed, for the same condition of loading. Three
such sets of moments are available from the tables in this Appendix:
(1) average moments in a square plate with a concentrated load at
the center, location (a). are given in Table 35 for four edges fixed
and in Table 36 for three edges fixed; (2) similar data for a load on
the diagonal at location (c) are also given in Tables 35 and 36;
(3) average moments are found in Table 38 for a centrally loaded
rectangular plate having b/a =0.5 and either four or three edges
fixed. The check of the carry-over factors is made as follows: First,
Edge
Condif-
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
consider a plate having four edges fixed. Then release the edge which
is simply-supported in the case with three edges fixed, and carry
over the average moments from that edge using the carry-over
factors from Table 1. Add these "carried-over" moments to those
already existing on the edges, and compare the total with the correct
average moments computed in this appendix for plates with only
three edges fixed.
0 0
+59/ +/160
-- t1 - -I /C
Values Given are --X/O 4
0
+6/4
-1614
-6/4
-/84
-798
(-835)
FIG. 22. CHECK OF CARRY-OVER FACTORS FOR PLATES
WITH CONCENTRATED LOADS
The calculations described above are illustrated in Fig. 22. The
quantity written below the line is the approximate average moment
in a plate having three edges fixed, as determined by means of the
distribution procedure using the carry-over factors derived for uni-
formly loaded plates. The moment in parentheses is the correct value
from the analyses described in the preceding sections. For a square
plate, the approximate procedure results in moments from 1 to 6
percent too low; while for the rectangular plate the range is from
120 ILLINOIS ENGINEERING EXPERIMENT STATION
4 percent low to 18 percent high. Although the agreement is not
exceptionally good, it is close enough to suggest that the distribu-
tion procedure developed for uniform loads might also be applicable
to plates carrying concentrated loads in the middle third of their area.
If concentrated loads are applied close to an edge, the distribu-
tion procedure proposed herein is not suitable, because of the rela-
tively large edge moments which are produced in the neighborhood
of the load. The concept of design for average moments, which
underlies the distribution procedure, is unsatisfactory if local effects
may be important as is the case for concentrated loads near an edge.
It should also be kept in mind that for a concentrated load at any
point on the plate, the local positive moments under the load are
usually much more critical than the moments at an edge.
MOMENTS IN TWO-WAY CONCRETE FLOOR SLABS
APPENDIX D
LIST OF SYMBOLS
The letter symbols and other notation used throughout this
bulletin are listed below:
x, y = horizontal rectangular coordinates having their
origin at the corner of a panel.
a = length of long span of slab panel.
b = length of short span of slab panel.
b/a = ratio of short span to long span.
t = total thickness of slab.
I = P/12 = moment of inertia per unit width of the
slab in a particular panel.
Ib = moment of inertia of the cross-section of a beam.
E = modulus of elasticity of the material in the slab.
Eb = modulus of elasticity of the material in a beam.
= Poisson's ratio of the material in the slab, taken
equal to zero throughout this bulletin.
N = EI(1 - U2) = measure of stiffness of an element of
the slab in a particular panel.
H = EbIb/bN or EbIb/aN = a dimensionless coefficient
which is a measure of the stiffness of a beam rela-
tive to that of the slab. The term b or a in the
denominator corresponds to the span length of the
beam being considered.
w = load per unit of area uniformly distributed over a
panel of the slab.
P = concentrated load applied to the slab.
Mi, My = bending moments per unit of width in the direc-
tion of x or y, respectively, acting on a section
normal to the x or y axis, respectively, positive
when producing compression at the top of the slab.
Mb = M, = bending moment acting on a section parallel
to the side of a panel having a length equal to b.
M° = M, = bending moment acting on a section parallel
to the side of a panel having a length equal to a.
Mtot = total bending moment acting on a section ex-
tending the full width of a panel in the direction
normal to that of the moment, expressed in ft-lb
or equivalent units.
M,. = average bending moment per unit of width, acting
on a section extending the full width of a panel,
ILLINOIS ENGINEERING EXPERIMENT STATION
expressed in ft-lb per foot of width, or simply in
pounds. May = Mtot/b or Mtot/a, depending on
the direction of the moment.
Ka, Kb = flexural stiffness of a panel of the slab at an edge
having a length of a or b, respectively.
ka, kb = dimensionless coefficients used to determine the
stiffness of a slab panel; Ka = kaN b and Kb =
kbN /b.
Cbb, Cba, Caa, Cab = carry-over factors for average moments acting on
the edges of a panel. The significance of the sub-
scripts is discussed in Section 6.
K', Cvb, etc. = modified stiffness and carry-over factors defined in
Section 7.
Fbb, Fba, Faa, Fab = positive moment correction factors employed to
determine the moments produced in the interior
of a panel by moments acting on the edges, de-
fined in Section 6.
4av = the average rotation of an edge of a panel.
G = Eb/[2(l+1) ] = modulus of elasticity in shear of
the material in a beam.
J = measure of the torsional rigidity of the cross-
section of a beam.
Tb, Ta = torsional stiffness of a beam having a span of b
or a, respectively, defined in Section 19.
T/K = a measure of the torsional stiffness of a beam rela-
tive to that of the slab along the edge adjacent
to the beam.
Symbols and expressions taken from the works of others and
referred to in this bulletin are not included in the foregoing list,
but are defined in the text where first introduced.
122