Lecture 37 Design of TwoWay Floor Slab System

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Lecture 37 - Design of Two-Way Floor Slab System

• December 2, 2002
• CVEN 444

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Lecture Goals

• Shear Strength of Slabs
• Shear Example
• Direct Design Method

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Shear Strength of Slabs
In two-way floor systems, the slab must have
adequate thickness to resist both bending moments
and shear forces at critical section. There are
three cases to look at for shear.
Two-way Slabs supported on beams Two-Way Slabs
without beams Shear Reinforcement in two-way
slabs without beams.
1. 2. 3.
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Shear Strength of Slabs
Two-way slabs supported on beams
The critical location is found at d distance from
the column, where
The supporting beams are stiff and are capable of
transmitting floor loads to the columns.
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Shear Strength of Slabs
The shear force is calculated using the
triangular and trapezoidal areas. If no shear
reinforcement is provided, the shear force at a
distance d from the beam must equal
where,
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Shear Strength of Slabs
Two-Way Slabs without beams
There are two types of shear that need to be
addressed
One-way shear or beam shear at distance d from
the column Two-way or punch out shear which
occurs along a truncated cone.
1. 2.
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Shear Strength of Slabs
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Shear Strength of Slabs
One-way shear considers critical section a
distance d from the column and the slab is
considered as a wide beam spanning between
supports.
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Shear Strength of Slabs
Two-way shear fails along a a truncated cone or
pyramid around the column. The critical section
is located d/2 from the column face, column
capital, or drop panel.
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Shear Strength of Slabs
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Shear Strength of Slabs
If shear reinforcement is not provided, the shear
strength of concrete is the smaller of
as is 40 for interior columns, 30 for edge
columns, and 20 for corner columns.
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Shear Strength of Slabs
Shear Reinforcement in two-way slabs without
beams.
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Shear Strength of Slabs
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Shear Strength of Slabs
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Shear Strength of Slabs
Conventional stirrup cages
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Shear Strength of Slabs
Studded steel strips
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Shear Strength of Slabs
The reinforced slab follows section 11.12.4 in
the ACI Code, where Vn can not
The spacing, s, can not exceed d/2. If a
shearhead reinforcement is provided
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Example Problem
Determine the shear reinforcement required for an
interior flat panel considering the following
Vu 195k, slab thickness 9 in., d 7.5 in., fc
3 ksi, fy 60 ksi, and column is 20 x 20 in.
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Example Problem
Compute the shear terms find b0 for
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Example Problem
Compute the maximum allowable shear Vu 195
k gt 135.6 k Shear reinforcement is need!
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Example Problem
Compute the maximum allowable shear So fVn
gtVu Can use shear reinforcement
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Example Problem
Use a shear head or studs as in inexpensive
spacing. Determine the a for
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Example Problem
Determine the a for The depth ad
41.8 in. 7.5 in. 49.3 in. ? 50 in.
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Example Problem
Determine shear reinforcement The fVs per
side is fVs / 4 14.85 k
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Example Problem
Determine shear reinforcement Use a 3 stirrup
Av 2(0.11 in2) 0.22 in2
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Example Problem
Determine shear reinforcement spacing Maximum
allowable spacing is
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Example Problem
Use s 3.5 in. The total distance is 15(3.5
in.) 52.5 in.
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Example Problem
The final result 15 stirrups at total distance
of 52.5 in. So that a 45 in. and c 20 in.
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Direct Design Method for Two-way Slab
Method of dividing total static moment Mo into
positive and negative moments.
Limitations on use of Direct Design method

• Minimum of 3 continuous spans in each direction.
  (3 x 3 panel)
• Rectangular panels with long span/short span
  2

1. 2.
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Direct Design Method for Two-way Slab
Limitations on use of Direct Design method
Successive span in each direction shall not
differ by more than 1/3 the longer span.
3. 4.
Columns may be offset from the basic rectangular
grid of the building by up to 0.1 times the span
parallel to the offset.
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Direct Design Method for Two-way Slab
Limitations on use of Direct Design method
All loads must be due to gravity only (N/A to
unbraced laterally loaded frames, from mats or
pre-stressed slabs) Service (unfactored) live
load 2 service dead load
5. 6.
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Direct Design Method for Two-way Slab
Limitations on use of Direct Design method

• For panels with beams between supports on all
• sides, relative stiffness of the beams in the 2
• perpendicular directions.
• Shall not be less than 0.2 nor greater than 5.0

7.
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Definition of Beam-to-Slab Stiffness Ratio, a
Accounts for stiffness effect of beams located
along slab edge reduces deflections
of panel adjacent to beams.
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Definition of Beam-to-Slab Stiffness Ratio, a
With width bounded laterally by centerline of
adjacent panels on each side of the beam.
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Two-Way Slab Design
Static Equilibrium of Two-Way Slabs
Analogy of two-way slab to plank and beam
floor Section A-A Moment per ft width in
planks Total Moment
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Two-Way Slab Design
Static Equilibrium of Two-Way Slabs
Analogy of two-way slab to plank and beam
floor Uniform load on each beam Moment in one
beam (Sec B-B)
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Two-Way Slab Design
Static Equilibrium of Two-Way Slabs
Total Moment in both beams Full load was
transferred east-west by the planks and then was
transferred north-south by the beams The same is
true for a two-way slab or any other floor system.
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Basic Steps in Two-way Slab Design

• Choose layout and type of slab.
• Choose slab thickness to control deflection.
  Also, check if thickness is adequate for shear.
• Choose Design method
• Equivalent Frame Method- use elastic frame
  analysis to compute positive and negative moments
• Direct Design Method - uses coefficients to
  compute positive and negative slab moments

1. 2. 3.
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Basic Steps in Two-way Slab Design

• Calculate positive and negative moments in the
  slab.
• Determine distribution of moments across the
  width of the slab. - Based on geometry and beam
  stiffness.
• Assign a portion of moment to beams, if present.
• Design reinforcement for moments from steps 5 and
  6.
• Check shear strengths at the columns

4. 5. 6. 7. 8.
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Minimum Slab Thickness for two-way construction
Maximum Spacing of Reinforcement At points of
max. /- M Min Reinforcement Requirements
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Distribution of Moments
Slab is considered to be a series of frames in
two directions
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Distribution of Moments
Slab is considered to be a series of frames in
two directions
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Distribution of Moments
Total static Moment, Mo
where
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Column Strips and Middle Strips
Moments vary across width of slab panel
Design moments are averaged over the width of
column strips over the columns middle strips
between column strips.
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Column Strips and Middle Strips
Column strips Design w/width on either side of a
column centerline equal to smaller of
l1 length of span in direction moments are being
determined. l2 length of span transverse to l1
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Column Strips and Middle Strips
Middle strips Design strip bounded by two column
strips.
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Positive and Negative Moments in Panels
M0 is divided into M and -M Rules given in ACI
sec. 13.6.3
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Moment Distribution
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Positive and Negative Moments in Panels
M0 is divided into M and -M Rules given in ACI
sec. 13.6.3
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Longitudinal Distribution of Moments in Slabs
For a typical interior panel, the total static
moment is divided into positive moment 0.35 Mo
and negative moment of 0.65 Mo. For an
exterior panel, the total static moment is
dependent on the type of reinforcement at the
outside edge.
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Distribution of M0
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Moment Distribution
The factored components of the moment for the
beam.
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Transverse Distribution of Moments
The longitudinal moment values mentioned are for
the entire width of the equivalent building
frame. The width of two half column strips and
two half-middle stripes of adjacent panels.
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Transverse Distribution of Moments
Transverse distribution of the longitudinal
moments to middle and column strips is a function
of the ratio of length l2/l1,a1, and bt.
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Transverse Distribution of Moments
Transverse distribution of the longitudinal
moments to middle and column strips is a function
of the ratio of length l2/l1,a1, and bt.
torsional constant
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Distribution of M0
ACI Sec 13.6.3.4 For spans framing into a common
support negative moment sections shall be
designed to resist the larger of the 2 interior
Mus ACI Sec. 13.6.3.5 Edge beams or edges of
slab shall be proportioned to resist in torsion
their share of exterior negative factored moments
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Factored Moment in Column Strip
Ratio of flexural stiffness of beam to stiffness
of slab in direction l1. Ratio of torsional
stiffness of edge beam to flexural stiffness of
slab(width to beam length)
a1
bt
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Factored Moment in an Interior Strip
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Factored Moment in an Exterior Panel
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Factored Moment in an Exterior Panel
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Factored Moment in Column Strip
Ratio of flexural stiffness of beam to stiffness
of slab in direction l1. Ratio of torsional
stiffness of edge beam to flexural stiffness of
slab(width to beam length)
a1
bt
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Factored Moment in Column Strip
Ratio of flexural stiffness of beam to stiffness
of slab in direction l1. Ratio of torsional
stiffness of edge beam to flexural stiffness of
slab(width to beam length)
a1
bt
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Factored Moment in Column Strip
a1
Ratio of flexural stiffness of beam to stiffness
of slab in direction l1. Ratio of torsional
stiffness of edge beam to flexural stiffness of
slab(width to beam length)
bt
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Factored Moments
Factored Moments in beams (ACI Sec. 13.6.3)
Resist a percentage of column strip moment plus
moments due to loads applied directly to beams.
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Factored Moments
Factored Moments in Middle strips (ACI Sec.
13.6.3)
The portion of the Mu and - Mu not resisted by
column strips shall be proportionately assigned
to corresponding half middle strips. Each middle
strip shall be proportioned to resist the sum of
the moments assigned to its 2 half middle strips.
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ACI Provisions for Effects of Pattern Loads
The maximum and minimum bending moments at the
critical sections are obtained by placing the
live load in specific patterns to produce the
extreme values. Placing the live load on all
spans will not produce either the maximum
positive or negative bending moments.
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ACI Provisions for Effects of Pattern Loads
The ratio of live to dead load. A high ratio
will increase the effect of pattern loadings. The
ratio of column to beam stiffness. A low ratio
will increase the effect of pattern
loadings. Pattern loadings. Maximum positive
moments within the spans are less affected by
pattern loadings.
1. 2. 3.
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Reinforcement Details Loads
After all percentages of the static moments in
the column and middle strip are determined, the
steel reinforcement can be calculated for
negative and positive moments in each strip.
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Reinforcement Details Loads
Calculate Ru and determine the steel ratio r,
where f 0.9. As rbd. Calculate the minimum
As from ACI codes. Figure 13.3.8 is used to
determine the minimum development length of the
bars.
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Minimum extension for reinforcement in slabs
without beams(Fig. 13.3.8)