حسن اياد وعلي حسين
Al-Khwarizmi
Engineering
Journal
Al-Khwarizmi Engineering Journal, Vol. 8, No. 3, PP 24 - 39 (2012)
Empirical Equations for Analysis of Two-Way Reinforced Concrete Slabs
Hassan Ayad Fadhil Ali Hussein Ali Al-Ahmed*
Department of Civil Engineering/ College of Engineering/ University of Baghdad
*Email: ali_hussein_alahmed@yahoo.co.uk
(Received 9 June 2011; accepted 7 August 2012)
Abstract
There are many different methods for analysis of two-way reinforced concrete slabs. The most efficient methods
depend on using certain factors given in different codes of reinforced concrete design. The other ways of analysis of
two-way slabs are the direct design method and the equivalent frame method. But these methods usually need a long
time for analysis of the slabs.
In this paper, a new simple method has been developed to analyze the two-way slabs by using simple empirical
formulae, and the results of final analysis of some examples have been compared with other different methods given in
different codes of practice.
The comparison proof that this simple proposed method gives good results and it can be used in analysis of two-way
slabs instead of other methods.
Keywords: Analysis, Two-way, Reinforced concrete slabs, Empirical equations, Inflection lines.
1. Introduction
There are many methods to estimate the values
of the bending moments occur in reinforced
concrete slabs, and perhaps the most common
methods which depend on coefficients taken from
special tables available in codes such as the
method of BS Code CP110 and the method of
ACI Code 63. These methods are approximate but
practical and were formed in such a way that the
moments are conservative because these methods
neglected many important factors to obtain
positive and negative bending moments by simple
and fast way without complexity. The high
accuracy in design calculations of structures is
undesirable because there is no capability of
estimating many factors affecting on design
results such as live loads, material properties and
methods of analysis and many other factors.
2. Analysis of Two Way Slab System
The coefficients used in different codes depend
on the aspect ratio of reinforced concrete slabs,
the boundary conditions at their edges (method of
restrained) and the continuity or discontinuity of
the edges. And to find negative and positive
bending moments, these coefficients are
multiplied by the load per unit area by the square
of the span. So it is important for the designer to
use tables to find these coefficients.
The most common manual calculation methods
for calculating bending moments in reinforced
concrete slabs are:-
1. Method two in ACI 63 Code (1): This
method is the most common method used in
design because of the simplicity in spite of that
this method gives high conservative results which
leads to increase in the quantity of steel
reinforcement.
2. Method three in ACI 63 Code (1): This
method is recommended to use by the latest ACI
codes because it is more accurate than method
two but it is more complicated.
3. Method one in ACI 63 Code (1): This
method is seldom used in spite of it is more
accurate than the above mentioned two methods
because it needs more effort calculations.
mailto:ali_hussein_alahmed@yahoo.co.uk
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
25
4. Method of BS CP110 (3 and 6).
5. ACI Direct design method (2, 4 and 5):
This method is limited so it cannot be used in
many cases.
6. Equivalent frame method: This method in
spite of it is accuracy and adequacy but it needs
much effort and time for calculations.
3. Description of the Proposed Method
The proposed method in this paper depends on
evaluating the coefficient (C) from simple
empirical equation. The numbers 0.26 & 0.67
mentioned below were predicted from curve
fitting to suit the other methods given by the ACI-
Code and CP110. This equation is expressed as:-
26.0
l
l
67.0C 1 −= … (1)
Where:-
l: is the distance between inflection lines in
direction of bending moment required.
l1: is the distance between inflection lines in
direction opposite to the direction of bending
moment required.
To obtain positive and negative bending
moments, the equation given below has been
developed
2SwBCM ×××= (as method 1 ACI- Code
63) … (2)
Where:-
B: is a coefficient extracted from ACI Code (1) or
BS CP110 (3 and 6) which is dedicated to find
flexural bending in beams or one-way slabs as
shown in Fig. (1).
w: is the total applied load (dead plus live) per
unit area.
S: is the clear span in the direction of the required
bending moment.
The identification of inflection lines between
the positive and negative bending moments
usually depends on approximate methods. It was
found that the location of these inflection lines
depends mainly on dimensions of spans of the
neighboring panels. Also, in case of continuous
panels from both sides, the ratio of
L
l
or (
1
1
L
l
) is
0.76. While, in the case of panel continuous from
one side and discontinuous from opposite side, the
ratio of
L
l
or (
1
1
L
l
) is 0.87.
Where:-
L: is the clear span in direction of required
bending moment (same as S).
L1: is the clear span in direction opposite to the
required bending moment
It is worth to mention that these values are
accurate and acceptable if the ratio of the spans of
the neighboring panels ranging between (2/3 -
3/2). Otherwise (when the ratio is beyond this
range) the identification of the inflection lines
may be evaluated according to the theory of
structures. Fig. (2) Shows the inflection lines in
panels according to ACI 63 Code (1).
Note: The values shown in Fig.(1) are not be
applicable if the larger of two adjacent spans is
greater than the shorter by 20%.
4. Comparison of Results with Other
Methods
A comparison study was done to check the
adequacy of the proposed method as it compared
with the results obtained using the ACI Code (1)
and the BS Standard methods (3). Tables (1, 2, 3
and 4) illustrate flexural positive and negative
bending moments as functions to the applied load
(w) for the proposed panels shown in Fig. (3, 4, 5
and 6) respectively. In these tables the symbols
(M+)x, (M+)y, (M-)x and (M-)y are defined as below:-
(M+)x = Maximum positive bending moment in
short direction.
(M-)x = Maximum negative bending moment in
short direction at continuous edges.
(M+)y = Maximum positive bending moment in
long direction.
(M-)y = Maximum negative bending moment in
long direction at continuous edges.
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
26
1
14
1
9
1
9
1
14
1
14
1
10
1
11
1
16
1
11
1
11
(a) Two Continuous Spans.
(b) More Than Two Continuous Spans.
Fig. 1. Values of the Coefficient (B) Used in Equation (2).
Beams Beam
L
(d) Panel Continuous at Three Edges.
l= 0.87L
1l= 0.76L1
L1
Inflection lines
(e) Panel Continuous at All Edges.
L1
1l= 0.76L1
L l= 0.76L Inflection lines
Fig. 2. Inflection Lines in Reinforced Concrete Panels According to ACI 63 Code.
L1= l1
L= l
(a) Panel Discontinuous at All Edges.
L1= l1
L
l= 0.87L Inflection line
(b) Panel Continuous at One Edge.
L
(c) Panel Continuous at Two Edges.
L1
l1= 0.87L1
l= 0.87L Inflection lines
L1= l1
L= l
(a) Panel Discontinuous at All Edges.
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
27
Fig. 3. Proposed Panels of m=0.6 Used for the Comparison between the Proposed Method and Other Methods.
Table 1,
Comparison among Different Methods.
Panel Moment BS Method
Method
(II)
ACI 63
Code
Method (III),
ACI 63 Code Average of
Methods
Standard
Deviation
Proposed
Method Dead
loads
Live
loads Average
I
(M+)x 0.786 0.765 0.687 0.868 0.778 0.776 0.00865 0.793
(M-)x 1.050 1.011 1.153 1.153 1.153 1.071 0.05990 1.110
(M+)y 0.454 0.480 0.252 0.324 0.288 0.407 0.08505 0.365
(M-)y 0.609 0.635 0.396 0.396 0.396 0.547 0.10706 0.511
II
(M+)x 0.605 0.674 0.467 0.765 0.616 0.632 0.03027 0.825
(M-)x 0.799 0.894 1.102 1.102 1.102 0.932 0.12653 1.200
(M+)y 0.363 0.402 0.144 0.252 0.198 0.321 0.08842 0.234
(M-)y 0.480 0.531 0.216 0.216 0.216 0.409 0.13805 0.328
III
(M+)x 0.566 0.609 0.441 0.752 0.597 0.591 0.01812 0.694
(M-)x 0.747 0.816 1.050 1.050 1.050 0.871 0.12967 1.009
(M+)y 0.311 0.324 0.144 0.252 0.198 0.278 0.05658 0.320
(M-)y 0.415 0.428 0.360 0.360 0.360 0.401 0.02947 0.465
IV
(M+)x 0.730 0.674 0.622 0.842 0.732 0.712 0.02688 0.662
(M-)x 0.955 0.894 1.037 1.037 1.037 0.962 0.05859 0.927
(M+)y 0.363 0.402 0.252 0.324 0.288 0.351 0.04731 0.450
(M-)y 0.480 0.531 0.648 0.648 0.648 0.553 0.07033 0.655
V
(M+)x 1.080 0.881 0.946 0.998 0.972 0.978 0.08134 1.153
(M-)x - - - - - - - -
(M+)y 0.557 0.570 0.432 0.396 0.414 0.514 0.07068 0.520
(M-)y 0.739 0.752 0.864 0.864 0.864 0.785 0.05611 0.727
VI
(M+)x 0.873 0.881 0.726 0.881 0.804 0.853 0.03457 0.947
(M-)x 1.149 1.166 1.231 1.231 1.231 1.182 0.03534 1.326
(M+)y 0.557 0.570 0.216 0.288 0.252 0.460 0.14694 0.404
(M-)y - - - - - - - -
x
y
6 m
3.6 m
m =
spanlong
spanshort
=0.6
6 m
6 m
6 m
3.6 m 3.6 m 3.6 m
Panel I
Panel V
Panel II
Panel IV Panel III
Panel VI
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
28
Table 2,
Comparison among Different Methods.
Panel Moment BS Method
Method
(II)
ACI 63
Code
Method (III),
ACI 63 Code Average of
Methods
Standard
Deviation
Proposed
Method Dead
loads
Live
loads Average
I
(M+)x 0.896 0.891 0.779 0.965 0.872 0.886 0.01034 0.851
(M-)x 1.200 1.173 1.339 1.339 1.339 1.237 0.07273 1.192
(M+)y 0.560 0.592 0.348 0.432 0.390 0.514 0.08865 0.480
(M-)y 0.752 0.784 0.588 0.588 0.588 0.708 0.08585 0.672
II
(M+)x 0.704 0.779 0.539 0.843 0.691 0.725 0.03878 0.890
(M-)x 0.912 1.029 1.317 1.317 1.317 1.086 0.17018 1.295
(M+)y 0.448 0.496 0.192 0.348 0.270 0.405 0.09722 0.335
(M-)y 0.592 0.656 0.324 0.324 0.324 0.524 0.14382 0.469
III
(M+)x 0.656 0.688 0.501 0.827 0.664 0.669 0.01360 0.745
(M-)x 0.848 0.923 1.216 1.216 1.216 0.996 0.15878 1.084
(M+)y 0.384 0.400 0.228 0.384 0.306 0.363 0.04106 0.420
(M-)y 0.512 0.528 0.540 0.540 0.540 0.527 0.01147 0.611
IV
(M+)x 0.816 0.779 0.683 0.917 0.800 0.798 0.01515 0.706
(M-)x 1.072 1.029 1.152 1.152 1.152 1.084 0.05097 0.989
(M+)y 0.448 0.496 0.348 0.432 0.390 0.445 0.04334 0.565
(M-)y 0.592 0.656 0.924 0.924 0.924 0.724 0.14382 0.822
V
(M+)x 1.216 1.024 1.003 1.083 1.043 1.094 0.08638 1.229
(M-)x - - - - - - - -
(M+)y 0.688 0.704 0.540 0.540 0.54 0.644 0.07383 0.652
(M-)y 0.912 0.928 1.200 1.200 1.200 1.013 0.13216 0.912
VI
(M+)x 1.024 1.024 0.848 1.003 0.926 0.991 0.04620 1.023
(M-)x 1.344 1.355 1.477 1.477 1.477 1.392 0.06027 1.432
(M+)y 0.688 0.704 0.276 0.396 0.336 0.576 0.16983 0.579
(M-)y - - - - - - - -
Fig. 4. Proposed Panels of m=0.667 Used for the Comparison between the Proposed Method and Other
Methods.
m =
spanlong
spanshort
=0.667
x
y 6 m
6 m
6 m
6 m
4.0 m 4.0 m 4.0 m 4.0 m
Panel I
Panel V
Panel II
Panel IV Panel III
Panel VI
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
29
Table 3,
Comparison among Different Methods.
Panel Moment BS Method
Method
(II)
ACI 63
Code
Method (III),
ACI 63 Code Average of
Methods
Standard
Deviation
Proposed
Method Dead
loads
Live
loads Average
I
(M+)x 0.998 1.007 0.852 1.039 0.946 0.984 0.02689 0.904
(M-)x 1.332 1.330 1.504 1.504 1.504 1.389 0.08156 1.265
(M+)y 0.678 0.716 0.444 0.552 0.498 0.631 0.09508 0.595
(M-)y 0.910 0.949 0.804 0.804 0.804 0.888 0.06127 0.833
II
(M+)x 0.780 0.871 0.613 0.916 0.765 0.805 0.04684 0.951
(M-)x 1.037 1.155 1.529 1.529 1.529 1.240 0.20973 1.383
(M+)y 0.542 0.600 0.240 0.444 0.342 0.495 0.11052 0.435
(M-)y 0.716 0.794 0.468 0.468 0.468 0.659 0.13899 0.609
III
(M+)x 0.734 0.761 0.555 0.897 0.726 0.740 0.01497 0.791
(M-)x 0.959 1.020 1.368 1.368 1.368 1.116 0.18016 1.150
(M+)y 0.465 0.484 0.300 0.480 0.390 0.446 0.04058 0.521
(M-)y 0.620 0.639 0.732 0.732 0.732 0.664 0.04894 0.757
IV
(M+)x 0.889 0.871 0.723 0.981 0.852 0.871 0.01511 0.744
(M-)x 1.177 1.155 1.226 1.226 1.226 1.186 0.02968 1.042
(M+)y 0.542 0.600 0.444 0.552 0.498 0.547 0.04177 0.681
(M-)y 0.716 0.794 1.212 1.212 1.212 0.907 0.21777 0.990
V
(M+)x 1.332 1.162 1.033 1.129 1.081 1.192 0.10460 1.294
(M-)x - - - - - - - -
(M+)y 0.832 0.852 0.684 0.684 0.684 0.789 0.07493 0.784
(M-)y 1.104 1.123 1.512 1.512 1.512 1.246 0.18802 1.097
VI
(M+)x 1.140 1.162 0.949 1.097 1.023 1.108 0.06101 1.093
(M-)x 1.521 1.536 1.723 1.723 1.723 1.593 0.09189 1.530
(M+)y 0.832 0.852 0.396 0.540 0.468 0.717 0.17649 0.754
(M-)y - - - - - - - -
Fig. 5. Proposed Panels of m=0.733 used for the Comparison between the Proposed Method and Other
Methods.
m =
spanlong
spanshort
=0.733
x
y 6 m
6 m
6 m
6 m
4.4 m 4.4 m 4.4 m 4.4 m
Panel I
Panel V
Panel II
Panel IV Panel III
Panel VI
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
30
Table 4,
Comparison among Different Methods.
Panel Moment BS Method
Method
(II)
ACI 63
Code
Method (III),
ACI 63 Code Average of
Methods
Standard
Deviation
Proposed
Method Dead
loads
Live
loads Average
I
(M+)x 1.083 1.106 0.899 1.106 1.003 1.064 0.04414 0.950
(M-)x 1.440 1.475 1.636 1.636 1.636 1.517 0.08535 1.331
(M+)y 0.806 0.852 0.576 0.720 0.648 0.769 0.08737 0.710
(M-)y 1.083 1.129 1.044 1.044 1.044 1.085 0.03474 0.994
II
(M+)x 0.864 0.945 0.668 0.968 0.818 0.876 0.05250 1.006
(M-)x 1.140 1.267 1.728 1.728 1.728 1.378 0.25263 1.463
(M+)y 0.645 0.714 0.360 0.612 0.486 0.615 0.09547 0.535
(M-)y 0.852 0.945 0.612 0.612 0.612 0.803 0.14029 0.750
III
(M+)x 0.783 0.829 0.599 0.945 0.772 0.795 0.02469 0.832
(M-)x 1.037 1.106 1.498 1.498 1.498 1.214 0.20302 1.210
(M+)y 0.553 0.576 0.396 0.612 0.504 0.544 0.03003 0.621
(M-)y 0.737 0.760 0.972 0.972 0.972 0.823 0.10578 0.903
IV
(M+)x 0.956 0.945 0.737 1.014 0.876 0.926 0.03541 0.776
(M-)x 1.256 1.267 1.267 1.267 1.267 1.263 0.00519 1.087
(M+)y 0.645 0.714 0.540 0.684 0.612 0.657 0.04250 0.796
(M-)y 0.852 0.945 1.476 1.476 1.476 1.091 0.27487 1.157
V
(M+)x 1.428 1.290 1.037 1.175 1.106 1.275 0.13190 1.350
(M-)x - - - - - - - -
(M+)y 0.991 1.014 0.792 0.828 0.810 0.938 0.09123 0.916
(M-)y 1.313 1.336 1.836 1.836 1.836 1.495 0.24131 1.282
VI
(M+)x 1.267 1.290 1.037 1.175 1.106 1.221 0.08186 1.156
(M-)x 1.693 1.705 1.981 1.981 1.981 1.793 0.13303 1.619
(M+)y 0.991 1.014 0.540 0.684 0.612 0.872 0.18432 0.928
(M-)y - - - - - - - -
Fig. 6. Proposed Panels of m=0.8 used for the Comparison between the Proposed Method and Other Methods.
m =
spanlong
spanshort
=0.8
x
y 6 m
6 m
6 m
6 m
4.8 m 4.8 m 4.8 m 4.8 m
Panel I
Panel V
Panel II
Panel IV Panel III
Panel VI
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
31
Figures (7, 8, 9, 10, 11 and 12) show the
relation between the aspect ratio (m) and the
bending moments as a function to (w) for panels
(I, II, III, IV, V and VI) respectively.
Fig. 7. Relationships between (m = short span/long span) and Bending Moments as Functions to w for Panel I.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
x -
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
x +
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
y +
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
y -
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
Continuous edges
Discontinuous edge
Mx-
Mx+
My+
My-
x
y
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
32
Fig. 8. Relationships between (m = short span/long span) and Bending Moments as Functions to w for panel II.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
y -
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
x +
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
x -
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
y +
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
Continuous edges
Discontinuous edge
Mx-
Mx+
My+
My- Continuous edge
Mx-
x
y
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
33
Fig. 9. Relationships between (m = short span/long span) and bending Moments as Functions to w for panel III.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
x +
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
x -
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
y +
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
y -
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
Continuous edges
Mx-
Mx+
My+
My-
Continuous edges
Mx-
My-
x
y
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
34
Fig. 10. Relationships between (m = short span/long span) and bending Moments as Functions to w for Panel IV.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
x +
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
x -
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
Mx+
My-
x
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
y +
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
y
-
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
Continuous edges
Mx-
My+
My-
Discontinuous edge y
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
35
Fig. 11. Relationships between (m = short span/long span) and Bending Moments as Functions to w for Panel V.
My+
x
Continuous edge My-
Discontinuous edges
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
x +
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
x -
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
y +
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.6 0.7 0.8
m
(M
y -
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
y
Mx+
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
36
Fig. 12. Relationships between (m = short span/long span) and Bending Moments as Functions to w for Panel VI.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
x +
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.6 0.7 0.8
m
(M
x -
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
y
+
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6 0.7 0.8
m
(M
y -
) a
s
a
re
la
tio
n
to
w
BS
Method II ACI-63
Method III ACI-63
Proposed method
Continuous edge
Mx-
Mx+
My+ Discontinuous edges
x
y
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
37
Example: Find the positive and negative flexural moments for panels I and II shown in Fig. (13):
Solution
Panel I
Flexural moments in short direction
472.026.0
87.04
76.05
67.0C =−
×
×
×=
w539.0w4
14
1
472.0M 2 =×××=+
w755.0w4
10
1
472.0M 2 =×××=−
Flexural moments in long direction
354.026.0
76.05
87.04
67.0C =−
×
×
×=
w552.0w5
16
1
354.0M 2 =×××=+
w804.0w5
11
1
354.0M 2 =×××=−
4 m
3 m
5 m
Panel I
Panel II
Fig. 13.
Panel II
Flexural moments in short direction
857.026.0
76.03
76.05
67.0C =−
×
×
×=
w482.0w3
16
1
857.0M 2 =×××=+
w701.0w3
11
1
857.0M 2 =×××=−
Flexural moments in long direction
142.026.0
76.05
76.03
67.0C =−
×
×
×=
w222.0w5
16
1
142.0M 2 =×××=+
w323.0w5
11
1
142.0M 2 =×××=−
Hassan Ayad Fadhil Al-Khwarizmi Engineering Journal, Vol. 8, No.3, PP 24- 39 (2012)
38
5. Discussion
1. In case of panel I (corner panel), the proposed
method gives less value of (M+)x & (M-)x,
(maximum difference is 14.10% & 18.64%
respectively) as compared with other methods
for large values of m. While (M+)y and (M-)y
are in good agreement with the other methods.
2. In case of panel II (discontinuous at one short
edge), the proposed method gives more value
of (M+)x as compared with the other methods
with maximum difference of 36.36%. While
(M-)x, (M+)y and (M-)y obtained by proposed
method are in good agreement with the other
methods.
3. In case of panel IV (discontinuous at one long
edge), the proposed method gives less value of
(M+)x & (M-)x (maximum difference is 18.83%
& 14.21% respectively) as compared with
other methods in all values of m. While (M+)y
and (M-)y are in good agreement with other
methods.
4. In case of panel VI (discontinuous at three
edges), the proposed method gives more value
of (M-)x for smaller value of m than the other
methods by maximum difference of 17.79% ,
while it gives less value of (M-)x for larger
value of m compared other methods with
maximum difference of 18.27%.
6. Recommendations
The proposed formulae presented in this paper
can be used for analyzing of reinforced concrete
two-way slabs supporting on beams. These slabs
are square or rectangular with aspect ratio not
exceeding 2. The coefficient B used in equation 2
can be taken from the coefficient used in ACI
code for evaluating the values of flexural moment
in beams or one-way slabs. If the difference
between two adjacent spans exceeds by more than
20% from the shorter span, then the value of
coefficient B can be estimated according to theory
of structures as well as the inflection lines.
7. References
[1] Building Code Requirements for Reinforced
Concrete (ACI-63). Detroit: American
Concrete Institute, 1963.
[2] Building Code Requirements for Reinforced
Concrete (ACI 318-11). Detroit: American
Concrete Institute, 2011.
[3] Code of Practice for the structural use of
Concrete, CP 110, Part 1, Nov. 1997.
[4] Wang C. K, Salmon C. G. and Pincheira J.
A., "Reinforced Concrete Design", 7th
edition, John Wiley and Sons, IWC, 2007.
[5] Nilson A. H., Darwin Da., and Dolan C. W.,
"Design of Concrete Structures", 14th
edition, McGrawhill Book Co., 2010.
[6] Allen A. H., "Reinforced Concrete Design to
CP 110", Cement and Concrete Association,
London, 1974.
)2012( 24 - 39 ، صفحة3دد، الع8 مجلة الخوارزمي الھندسیة المجلد حسن ایاد فاضل
39
معادالت وضعیة لتحلیل البالطات الخرسانیة المسلحة العاملة باتجاھین
*أحمد-علي حسین علي ال فاضل حسن ایاد
جامعة بغداد / كلیة الھندسة / قسم الھندسة المدنیة
ali_hussein_alahmed@yahoo.co.uk :االلكتروني البرید*
الخالصة
ان اغلب ھذه الطرق تعتمد اما على استخدام ثوابت محددة . تتوفر عدة طرق مثبتة في المراجع والمدونات لتحلیل البالطات الخرسانیة العاملة باتجاھین
او تعتم د طریق ھ . ف ي الم دونات الش ائعة االس تخدام مت وفرة وھناك جداول لھذه الثوابت. ابعاد البالطة وطریقة تثبیتھا عند الحافات ولحاالت مختلفة من نسبة
.التحلیل على طریقھ التصمیم المباشر او طریقة الھیكل المكافئ و ھذه الطریقتین تحتاج على االغلب الى وقت طویل لتحلیل البالطات
وللتاكد من صالحیة ھذه المعادالت تم اجراء دراسة . المعادالت للبالطات العاملة باتجاھین تم استنباط صیغ لمعادالت ریاضیة مبسطة والجل تبسیط حل
جاھین وقد تم عاملة باتمقارنة للنتائج المستحصلة من ھذه المعادالت مع الطرق المعتمدة في المدونات الشائعة االستخدام والمتبعة حالیا في تحلیل البالطات ال
ون الحاجھ الى الحصول على نتائج جیده من ھذه المقارنھ مما یفسح المجال لالستفادة من ھذه المعادالت في تحلیل البالطات باتجاھین بیسر وبوقت قصیر د
.الرجوع الى الجداول الموجودة في المدونات
mailto:ali_hussein_alahmed@yahoo.co.uk