Reinforced Concrete Design II – ACI 318

Lecture 5
Design of two way slabs
 Regula
(3

y
Introduction Plate/Shell (2D) z
x z x
A slab is a structural element whose thickness is small compared
t<<(x,z) to
its own length and width.
h
t  L , S zS
t t
Lx
Slabs in buildings are usually used to transmit the loads on floors and
roofs to the supporting beams Loads
Dimensional Hierarchy of Structural

Beam Beam Column

Slab Beam

Column Beam Beam

Footing
Slab

Beam Beam
Soil

2
 Types of Slabs

Solid slabs :- which are divided into

- One way solid slabs
- Two way solid slabs

Ribbed slabs :- which are divided into

- One way ribbed slabs
- Two way ribbed slabs One-way slab

Two-way slab

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 Solid Slab

L L
Two way slab 2 One-way slab 2
S S
Ribbed Slab (joist construction)

4 Two way slab One-way slab

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 Types of
two-way
Slabs

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7 Solid slab
8 Ribbed Slab
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10 Grid or Waffle slab
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 Grid or Waffle slab
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13
 Flat slab
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 Comparison of one-way and two-way Slab behavior economic
choices

• Flate plate (for relatively light loads as in appartments

or offices) suitable span 4.5 m to 6.0 m with LL=3 to 5
kN/m2.
o Advantages : low cost formwork, exposed flat
ceilings, fast.
o Disadvantages : Low shear capacity, low
stiffness (notable deflection)

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 Comparison of one-way and two-way Slab behavior economic
choices

• Flate slab (for heavy industrial loads) suitable span 6 m

to 9.0 m with LL=5 to 7 kN/m2.
o Advantages : low cost formwork, exposed flat
ceilings, fast.
o Disadvantages : need more formwork for
capital.

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 Comparison of one-way and two-way Slab behavior economic
choices

• Waffle slab (two-way joist system) suitable span 7.5 m

to 12.0 m with LL=4 to 7.5 kN/m2.
o Advantages : Carries heavy loads, attractive
exposed ceilings, fast.
o Disadvantages : High cost formwork

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 Comparison of one-way and two-way Slab behavior economic
choices

• One-way slab on beams suitable span 3 m to 6.0 m

with LL=3 to 5 kN/m2.
o Can be used for larger spans with relatively
higher cost and higher deflections
• One-way joist system suitable span 6.0 m to 9.0 m
with LL=4 to 6 kN/m2.
o Deep ribs, the concrete and steel quantities are
relative low, expensive formwork expected.

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 Behavior of slabs loaded to failure in flexure

For a two-way slab loaded

to failure in flexure, there
are four or more stages of
behavior

Inelastic action in a slab fixed on four sides.

 Two Way Slab Design by Direct Design Method

The ACI Code (13.5.1.1) specifies two methods for

designing two-way slabs for gravity loads:

1. Direct design method : that could have been named the

direct analysis method because it determines or
prescribes moments for different parts of the slab panel
without the need to conduct structural analysis, is
explained in the following sections.

2. Equivalent frame method.

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 Two Way Slab Design by Direct Design Method

• Beam-to-Slab Stiffness Ratio,

The ACI Code (13.5.1.1) specifies two methods for

Because the lengths l , of the beam and slab are equal,

this quantity is simplified and expressed in the code as

where Ecb and Ecs are the moduli of elasticity of the

beam concrete and slab concrete, respectively, and Ib
and Is are the moments of inertia of the uncracked
beams and slabs.

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 Two Way Slab Design by Direct Design Method

• Beam-to-Slab Stiffness Ratio,

 Beam and slab sections for

calculations of

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 Two Way Slab Design by Direct Design Method

• Beam-to-Slab Stiffness Ratio,

 Cross section of beams as defined in ACI Code

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 Two Way Slab Design by Direct Design Method

• Minimum Thickness of Two-Way Slabs

 Slabs without Beams between Interior Columns
For a slab without beams between interior columns and having a
ratio of long to short spans of 2 or less, the minimum thickness
is as given in Table 13-1 but is not less than 12.7 cm (5 in).

275 mPa
414 mPa
517 mPa

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 Two Way Slab Design by Direct Design Method

• Minimum Thickness of Two-Way Slabs

 Slabs with Beams between the Interior Supports

For slabs with beams between interior supports, ACI Code

Section 9.5.3.3 gives the following minimum thicknesses:

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 Two Way Slab Design by Direct Design Method
• Distribution of Moments within Panels—Slabs without
Beams between All Supports

 Statical
Moment, M0
For design, the slab
is considered to be
a series of frames in
the two directions,
as shown in Figure.

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 Two Way Slab Design by Direct Design Method
• Column and Middle Strips:

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 Two Way Slab Design by Direct Design Method
• Moment in (longitudinal) direction of bending considered:

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 Two Way Slab Design by Direct Design Method
• Determination of the total factored static moment:

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 Two Way Slab Design by Direct Design Method
• Longitudinal Distribution of Moments in Slabs
Interior span

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 Two Way Slab Design by Direct Design Method
• Longitudinal Distribution of Moments in Slabs
Exterior span

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 Two Way Slab Design by Direct Design Method
• Longitudinal Distribution of Moments in Slabs

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 Two Way Slab Design by Direct Design Method
• Transverse Distribution of Moments (to the column strips)

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 Two Way Slab Design by Direct Design Method
• Transverse Distribution of Moments (to the column strips)

The cross section is to be divided into separate rectangular

parts. Carry out the summation in such a way to give the largest
value of C, as shown in the figure.
x is the shorter dimension of the rectangular part of the cross
section and y is the longer dimension of the rectangular part of
the cross section.
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 Two Way Slab Design by Direct Design Method
• Transverse Distribution of Moments (to the column strips)
Interior negative moment

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 Two Way Slab Design by Direct Design Method
• Transverse Distribution of Moments (to the column strips)
Exterior negative moment

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 Two Way Slab Design by Direct Design Method
• Transverse Distribution of Moments (to the column strips)
Positive moment

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 Two Way Slab Design by Direct Design Method
• Transverse Distribution of Moments (to the beam)

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 Two Way Slab Design by Direct Design Method
• Design of Beams for Shear

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 Two Way Slab Design by Direct Design Method
• Design of Beams for Shear

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 Two Way Slab Design by Direct Design Method
• Minimum Reinforcement Requirements

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 Two Way Slab Design by Direct Design Method
• Minimum Reinforcement Requirements

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 Two Way Slab Design by Direct Design Method
• Maximum Spacing of Reinforcement

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 Two Way Slab Design by Direct Design Method
• Example
Design the interior strip shown for the structural plan using
DDM. The column dimensions are all 30cmX30cm. The
slab depth is 15 cm, and the beam depth is 50 cm. Ultimate
distributed load over the slab = 15 kN/m2.

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 Two Way Slab Design by DDM
• Solution :

Middle Panels
1- Determine strip size and dimensions:
l1=6.0 m, l2=6.5 m, l2‘=6.8 m

The 6.5 side:

l2/2=3.25m
min(l1/4=1.5m, l2/4=1.63m)=1.5 m
The 6.8 side:
l2’/2=3.4m
min(l1/4=1.5m, l2’/4=1.7m)=1.5 m
l1=6.0 m
ln=6.0 – 0.3 = 5.7 m
l2=(6.8+6.5)/2 = 6.65 m

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 Two Way Slab Design by DDM
• Solution :
Middle Panels

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 Two Way Slab Design by DDM
• Solution :
Middle Panels
3- Determine the total static moment:

4- Distribute the total static moment in the longitudinal direction:

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 Two Way Slab Design by DDM
• Solution :
Middle Panels
5- Distribute the interior negative moment to the column and
middle strips:

Linear interpolations shall be made between values 1 and 2 :

Coef. = 0.71751

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 Two Way Slab Design by DDM
• Solution :
Middle Panels
6- Distribute the positive moment to the column and middle strips:

Linear interpolations shall be made between values 1 and 2 :

Coef. = 0.71751

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 Two Way Slab Design by DDM
• Solution :
Middle Panels
7- Distribute the column strip moment to the slab and the beam:

85 % of the moment in the column strip is assigned to the

beam and the balance of 15 % is assigned to the slab in the
column strip.

Interior negative Positive

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 Two Way Slab Design by DDM
• Solution :
Middle Panels
Results

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 Two Way Slab Design by DDM
• Solution :
Middle Panels
Results

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 Two Way Slab Design by DDM
• Solution :
Exterior Panels
1- Determine strip size and dimensions:

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 Two Way Slab Design by DDM
• Solution :
Exterior Panels

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 Two Way Slab Design by DDM
• Solution :
Exterior Panels

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 Two Way Slab Design by DDM
• Solution :
Exterior Panels

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 Two Way Slab Design by DDM
• Solution :
Exterior Panels

Linear interpolations shall be made between values 1 and 2 :

Coef. = 0.728

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 Two Way Slab Design by DDM
• Solution :
Exterior Panels

Linear interpolations shall be made between values 1 and 2 :

Coef. = 0.728

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 Two Way Slab Design by DDM
• Solution :
Exterior Panels

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 Two Way Slab Design by DDM
• Solution :
Results

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