Perhitungan sturktur kolom baja 6162 (4).ppt

-1-
457.306
Theory of Reinforced Concrete and Lab. II
Fall 2009
10. Analysis and Design of Slabs
TYPES OF SLABS
DESIGN OF ONE-WAY SLABS
TEMPERATURE & SHRINKAGE REINFORCEMENT
TWO-WAY EDGE-SUPPORTED SLABS
TWO-WAY COLUMN-SUPPORTED SLABS
by Prof. Jae-Yeol Cho

-2-
- Slabs which are supported on two opposite side only
- Slabs of which the ratio of length to width is larger than
about 2
TYPES OF SLABS
One-way Slabs
Theory of Reinforced Concrete and Lab II. Fall 2009
Beam
Beam

-3-
TYPES OF SLABS
Two-way Slabs
Beam
Beam
Beam
Beam
<Two-way slab> <Flat plate>
<Flat slab> <Grid or waffle slab>

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Cylindrical bending : A one-way slab consists of a set of
parallel rectangular beams(strips).
 Design and Analysis of a one-way slab is almost the same
as those of a rectangular beam.
Simple supports
on two long
edges only
Main
reinforcement

-5-
Minimum Thickness h
KCI Code 4.3.1 specifies the minimum thickness for non-
prestressed slab of normal-weight concrete (wc=2,300kg/m3)
using 400MPa reinforcement.
Simply supported
One end continuous
Both end continuous
Cantilever
/ 28
l
/10
l
/ 20
l
/ 24
l

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Minimum Thickness h
Note
1) Span length l is in mm.
2) For light-weight concrete 1,500~2,000kg/m3, the values
shall be multiplied by (1.65-0.00031wc) but not less than
1.09, where wc is in kg/m3.
3) For fy other than 400MPa, the values shall be multiplied by
(0.43+fy /700).

-7-
Other Details
• The concrete protection below the reinforcement should
follow the requirements of KCI Code 5.4, calling for 20mm.
(25mm below the center of steel)
• The lateral spacing of the bars should not exceed 3 times
the thickness h or 400mm, whichever is less.
• Actual spacing is not less than about 1.5 times the slab
thickness to avoid excessive cost.
D10 or D13
25mm 20mm min.

-8-
• In one way slabs, it is necessary to provide special
reinforcement for shrinkage and temperature in the
direction perpendicular to the main reinforcement.
; know as temperature or shrinkage reinforcement
distribution steel.

-9-
Slabs where fy≤400MPa deformed bars are used
Slabs where fy≥400MPa at yield strain of 0.0035 is used
KCI Code provisions (5.7.2)
Specifies the minimum ratios of steel area to gross concrete
area.
But in no cases may them be placed farther apart than 5
times the slab thickness or more than 400mm.
In no case is the reinforcement ratio to be less than 0.0014.
0.002
0.002 400 / y
f


-10-
10. Analysis and Design of slab
Example 13.1 One –way Slab Design
A reinforced concrete slab is built integrally with its supports
and consists of two equal spans, each with a clear span of
4.5m.
The service live load is 5kN/m2 and 27MPa concrete is
specified for use with a yield stress equal to 400MPa.
Design the slab

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Solution>
- Determination of slab thickness
This structural system corresponds to the case of both ends
being continuous
160
28
l
mm

The trial thickness of 180mm will be need, for which the
weight is
   
3
2 2
.
180 10 24 4.32
unit weight conc
kN kN
m m

  

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- Factored load
2
: 1.2 4.32 5.184
: 1.6 5.00 8.0
13.184
DL
LL
kN
Total
m
 
 

- Factored moment at critical sections (Handout 13-1)
At interior support
At midspan
At exterior support
2
1
13.184 4.5 29.7
9
M kN m
     
2
1
13.184 4.5 19.1
14
M kN m
     
2
1
13.184 4.5 11.1
24
M kN m
     

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- The maximum reinforcement ratio (Handout #3-2)
 
2
max
27 0.003
0.85 0.021
400 0.003 0.004
  

- If the maximum ρ were actually used, the minimum required
effective depth, controlled by negative moment at the
interior support would be obtained
2
1 0.59
u
M
d
f
y
f b
y f
ck
 

 
 

 
 

-14-
2
2
2
1 0.59
n s y
y
y
ck
a
M A f d
f
f bd
f
Rbd
 



 
 
 
 
 
 
 
 

Recall
   
    
3 3
2
29.7 10 10
400
0.85 0.021 400 1000 1 0.59 0.021
27
5095mm

 
  
 
 

Homework #4 Complete the design

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• Consider a simplest type of TWO-WAY slab which is
supported along its four edges by relatively DEEP, STIFF,
MONOLITHIC concrete beams or by walls or steel griders.
The deflections at the
intersection point must be
the same.
where wa is the share of the distributed load w carried in
the short direction and wb is the share in the long direction.
4 4
5 5
384 384
a a b b
w l w l
EI EI

4
4
a b
b a
w l
w l
 a b
w w


Simple supports
on all four edges

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• The outer strips s2 and l2
are not only bent but also
TWISTED.
 This twisting results in
torsional stresses and
moments.
 Total loads on the slab is
carried by two-way
bending moment plus
twisting moments.

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Proof
Consider a simply supported square slab.
¡) assuming that only bending is present, the maximum
moment
¡¡) The exact theory of elastic plates gives 0.048wl2
 Twisting moments relieve the bending moment by about 25%
2
2
( / 2)
0.0625
8
w l
wl


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• If the load is increased, so that the steel at the middle of
strip s1 is yielding, the slab will not show immediate failure
due to inelastic redistribution.
 slabs need not be designed for the absolute maximum
moment in each of the two directions, but only for a
SMALLER AVERAGE moment in each of the two
directions in the central portion of the slab.
Note
For example, one of the several analytical methods in
general use permits a square slab to be designed for a
moment of 0.036wl2.
Center -> in the both direction

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• Most practical approximated design method considering the
variation in maximum moment is designing for a reduced
(averaged) moment in the outer quarters of the slab span
in each direction.
Ma along 1-1
Variation of Ma,max
across 2-2
Variation of Mb,max
across 1-1
Mb along 2-2

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Note
1) Two layers positive bars are placed in two-way slab.
 Short / Long direction bars are placed on the top of bars.
2) According to KCI Code 10.6.1, the minimum reinforcement
in each direction for two-way slabs is the same required for
shrinkage and temperature crack control as for one-way
slab.
3) The spacing of flexural reinforcement at critical section
must not exceed 2 times the slab thickness h.

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• The twisting moments are usually of importance only at
EXTERIOR corners, where they tend to crack the slab
- at the bottom along the panel diagonal
- at the top perpendicular to the panel diagonal
longer clear span
Top bars
Bottom bars

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• Consider a two-way slab which are supported by relatively
SHALLOW, FLEXIBLE beams, or flat plates, flat slabs, or
two-way joist system.
For column supported construction, 100% of the applied
load must be carried in each direction, jointly by the slab
and its supporting beams.
Effective beam Effective beam

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• Consider a flat floor supported by 4 columns,
assuming that l2 >l1
<critical moment sections> <moment variation along a span>

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In any span of a continuous beam, the sum of the midspan
positive moment and negative moments at adjacent supports
is equal to the midspan positive moment of a corresponding
simply supported beam.
Similarly,
 BUT, these results disclose NOTHING about the support
moments and midspan moments.
2
2 1
1 1
( ) ( )
2 8
ab cd ef
M M M wl l
  
2
1 2
1 1
( ) ( )
2 8
ac bd gh
M M M wl l
   <moment variation along a span>

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The moments across the width of critical sections, such as AB
or EF are NOT constant but vary as above.
 For design purpose, it is convenient to divide each panel
into column strips and middle strip between column strips.
<moment variation
along a span>
<critical moment sections> <moment variation across
the width of critical sections>
Variation assumed
for design
Actual moment
across AB
Actual
moment
across EF

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• KCI Code 10.3.2 permits design “by any procedure
satisfying conditions of equilibrium and geometrical
compatibility”.
In addition, specific reference is made to two alternative
approaches :
- a semi-empirical “direct design method (DDM)”
- an approximate elastic analysis known as the equivalent
frame method (EFM).

-27-
Note
1. Both DDM and EFM employ the concepts of column/middle
strip.
- A column strip has a width on each side of the column
centerline equal to one-fourth the smaller of the panel
dimensions.
2. Portions of slab to be included with beam.
<symmetric slab> <single side slab>

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