Two way slab

Design Of Reinforced Concrete Structures ii
Two-Way Slabs
1
1. Inroduction
When the ratio (L/S) is less than 2.0, slab is called two-way slab, as shown in the fig.
below. Bending will take place in the two directions in a dish-like form.
Accordingly, main reinforcement is required in the two directions.

Two-Way Slabs
2
2.Types Of Two Way Slabs
3. Design Methods
Two-way slabs
Slabs without
beams
Flat plates
Flat slabs
Slabs with beams
Two-way edge-
supported slab
Two-way ribbed
slab
Waffle slabs
Two-way Edge-
supported ribbed
slabs
Design methods
Simplified
Design Methods
Grashoff
method
Marcus method
Egyptian Code
method
Direct Design
Method"DDM"
Equivalent Frame
Method " EFM "

Two-Way Slabs
3
4. Direct Design Method "D.D.M"
Before discussion of this Method, we have to study some concepts:
1. Limitations:
1. Three or more spans in each direction.
2. Variation in successive spans 33% ( .
3. LL 2 DL
4. Column offset 10% in each direction.
5. L/B 2.
6. For slabs on beams, for one panel
.
2. Determination of Two way slab thickness:
 Case 1 : interior and edge beams are exist.
=
=
Where:
: is the largest clear distance in the longest direction of
panels.
: is the clear distance in the short direction in the panel.
=
=
Example for finding : for fig. shown:
For panel 1 …
For panel 5 …
So h to be used should be : hmin< h < hmax

Two-Way Slabs
4
 Case 2: interior beams are not existing, thickness can be found according to table 8.8,
page 339.
3. Estimating dimensions of interior and exterior beams sections:
Dimensions can be estimated from the following figures:
Where: b = beam width,
h = slab thickness,
a =beam thickness.

Two-Way Slabs
5
► Design Procedures
Discussion will be done to one representative strip in the horizontal and vertical
directions; the same procedure can be used for the other strips.
a- Determination of total factored Static Moment :
= Strip width /8
: total factored load in t/m2
.
= clear distance in the direction of strip, and not less than 0.65 .

Two-Way Slabs
6
b- Distribution of the total factored static moment to negative and positive moments:
I. For interior Spans:
According to the code, the moments can be distributed according to factores shown in
the figure:
II. For Edge Spans :
Static Mom. Mo can be distributed, according to factors given in the table 8.9, page 341.

Two-Way Slabs
7
c- Distribution of the positive and negative factored moments to the Column and middle
strips:
Note: width of column strip is equal to 0.25l1 or 0.25l2 which is smaller.
l1: length in the direction of strip, center to center between columns.
l2: length in the direction perpendicular to l1.
I. Determination of factored moments on column and middle strips:
 Finding α and βt:
 α =
α : is ratio of flexural stiffness .
Ib : Moment of inertia of the beam in the direction of strip… can be found from
fig.8.14 and fig.8.15, pages 310 and 311.
Is : Moment of inertia of slab = , where is slab thickness.
 βt = ,
βt: Ratio of torsional stiffness
and are the modulus of elastisity of concrete for beam and slab.
Note: βt is given only for edge
beams perpendicular to the strip
Note: α is given only for the
beams in the direction of the strip

Two-Way Slabs
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C: Cross sectional constant defines torsional properties C =
X: smallest dimension in the section of edge beam.
Y: Largest dimension in the section of edge beam.
Note: the C relation is applicable directly for rectangular section only, but when used for
L-Shape beams, we should divide it to two rectangular sections and find C.
C "A" = C1 + C2 for A and C "B" = C1 + C2 for B.
C to be used = Max (C "A" , C "B" ).
When α and βt are found, factors for moment can be found from table 8.10 page 343 for
the column strip.
Notes:
 α l2/l1 = 0.0 , when there is no interior beams in the direction of strip under
consideration.
 βt = 0.0 , when there is no extirior “edge” beams perpendicular to the strip under
consideration.

Two-Way Slabs
9
After finding the moments on the column strip, Moments on the middle strip is the
remain.
II. For the moment on the beam “ if exist ” :
If: α l2/l1 ≥ 1 … The beam moment is 85% of the moment of the column strip.
α l2/l1 = 0 … there is no beam .. mom. = 0
0 < α l2/l1 < 1 … Interpolation have to be done between 0 and 85% to find percentage
of moment on the beam from that of the column strip.
** The Mom. on the remain part of column strip = Tot. Mom. on the column strip –
Mom. on the beam.
Summary:
1- Find Mo :
2- Distribute M0 into +ve and –ve Mom.
3- Distribute Mom. Into column strip and Middle Strip.
Column strip Middle Strip
4- Distribute Mom. In column strip into Mom. On beam and remained slab.
On beam On remained Slab
After calculating Moments, we can find the ρ, then Ast required

Two-Way Slabs
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Example 1:
For the given data, design strip 1-2-3-4 of the two way slab for flexure.
Data:
Columns are 30cm X 30cm, Equivalent partitions load=250 Kg/m2
, Live Load = 400Kg/m2
,
= 280 kg/cm2
= 4200 Kg/cm2
, slab thickness = 16cm

Two-Way Slabs
11
Solution:
Thickness is given 16cm, no need to be checked.
1- Calculate total factored load Wu "t/m2
":
Wu = 1.4 (0.16 2.5 + 0.25) + 1.7 (0.4) = 1.59 t/m2
.
2- Determine The Total Factored Static Moment (Mo) :
Mo = = = 13.83t.m

Two-Way Slabs
12
3- Distribute Mo into +ve and –ve moments :
The total factored static moment was distributed according to Table "8.9" in your
text book as shown in the following Figure.

Two-Way Slabs
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4- Moments on the column Strip :
Evaluate the constant and
 Evaluation of :
α = . "For beam in direction of strip"
For a/h=50/16=3.125 and b/h=30/16=1.875, f=1.4 (Fig. 8.14)
α = = 4.07
 Evaluation of :
βt = . " For edge beam perpendicular to direction of strip"
CA = = 208,909.55
CB = = 128,532.48
C = Max (CA or CB) = 208,909.55
βt = = 0.97
After α and βt are calculated, factors for the moment of column strip can be found
from Table 8.10, page 343
α = 4.07 =4.88 1 , βt between 0 and 2.5 , = 1.2
 – Ve exterior moment Factor :
=1 =1.2 =2
βt = 0 100 100 100
βt = 0.97 0.903 0.8797 0.7866
βt = 2.5 75 69 45

Two-Way Slabs
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 +Ve interior moment Factor :
=1 =1.2 =2
75 69 45
 -Ve interior moment Factor :
=1 =1.2 =2
75 69 45

Two-Way Slabs
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5- Moments on the Middle Strip: "The remain moment":
6- Moment On Beam :
As α >1 …. Beam willl resist 85% of the column strip moment.
7- Moment On Remained Slab :

Two-Way Slabs
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By:
Eng. AbdUlla Taisir Al-Madhoun
Eng. Nour Nagi Al-Hindi
.
Notes:
 For each value of moment, can be calculated, then .
 Widths to used for design and ρ calculations are :
-For the remained slab of column strip: b = 1.25-0.64=0.61m
-For half middle strip: b= 3.15-1.25=1.9m
-Beam is designed as rectangular section of b=0.3m at –ve
moments and as T-section at +ve moments.
 Beam should be designed for shear, according to
specifications of code ACI 318"13.6.8", and reported in page
344 of your text book.

Two way slab

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Two way slab