230 Reinforced Concrete Design
Figure 8.4
Continuous slab
Beam
Beam
Beam
Beam
Beam
Span Span Span Span
7m
Plan
4.5m 4.5m 4.5m 4.5m
Elevation
Estimate of slab depth
As the end span is more critical than the interior spans, try a basic span–effective depth
ratio 30 per cent above the end-span limit of 26.0 (i.e. 33.0):
span
minimum effective depth ¼
33:0 correction factor
4500 136
¼ ¼
33:0 c.f. c.f.
As high yield steel is being used and the span is less than 7 m the correction factor can be
taken as unity. Try an effective depth of 140 mm. For a class XC-1 exposure the
cover ¼ 25 mm. Allowing, say, 5 mm as half the bar diameter of the reinforcing bar:
overall depth of slab ¼ 140 þ 25 þ 5 ¼ 170 mm
Slab loading
3
self-weight of slab ¼ 170 25 10 ¼ 4:25 kN/m2
total permanent load ¼ 1:0 þ 4:25 ¼ 5:25 kN/m2
For a 1 m width of slab
ultimate load, F ¼ ð1:35gk þ 1:5qk Þ4:5
¼ ð1:35 5:25 þ 1:5 3:0Þ4:5 ¼ 52:14 kN
Using the coefÞcients of table 8.1, assuming the end support is pinned, the moment at
the middle of the end span is given by
M ¼ 0:086Fl ¼ 0:086 52:14 4:5 ¼ 20:18 kN m
Bending reinforcement
M 20:18 106
¼ ¼ 0:0412
bd fck 1000 1402 25
2
From the lever-arm curve of Þgure 4.5, la ¼ 0:96. Therefore, lever-arm z ¼ la d ¼
0:95 140 ¼ 133 mm:
M 20:18 106
As ¼ ¼ ¼ 349 mm2 /m
0:87fyk z 0:87 500 133
Provide H10 bars at 200 mm centres, As ¼ 393 mm2 /m.
Design of reinforced concrete slabs 231
Check span–effective depth ratio
100As; req 100 349
¼ ¼ 0:249
bd 1000 140
From Þgure 6.3 this corresponds to a basic span–effective depth ratio in excess of
32 1:3 (for an end span) ¼ 41. The actual ratio ¼ 4500=140 ¼ 32:1; hence the
chosen effective depth is acceptable.
Similar calculations for the supports and the interior span give the steel areas shown
in Þgure 8.5.
H10 – 400
Figure 8.5
H10 – 400 H10 – 200 H10 – 250
Reinforcement in a
continuous slab
H10 – 200 H10 – 400 H10 – 250 H10 – 250
At the end supports there is a monolithic connection between the slab and the beam,
therefore top steel should be provided to resist any negative moment. The moment to be
designed for is a minimum of 25 per cent of the span moment, that is 5.1 kN m. In fact, to
provide a minimum of 0.13 per cent of steel, H10 bars at 400 mm centres have been
speciÞed. The layout of the reinforcement in Þgure 8.5 is according to the simpliÞed
rules for curtailment of bars in slabs as illustrated in Þgure 8.2.
Transverse reinforcement ¼ 0:0013bd
¼ 0:0013 1000 140
¼ 182 mm2 /m
Provide H10 at 400 mm centres top and bottom, wherever there is main reinforcement
(196 mm2/m).
8.5 Solid slabs spanning in two directions
When a slab is supported on all four of its sides it effectively spans in both directions,
and it is sometimes more economical to design the slab on this basis. The amount of
bending in each direction will depend on the ratio of the two spans and the conditions
of restraint at each support.
If the slab is square and the restraints are similar along the four sides then the load
will span equally in both directions. If the slab is rectangular then more than one-half of
the load will be carried in the stiffer, shorter direction and less in the longer direction. If
one span is much longer than the other, a large proportion of the load will be carried in
the short direction and the slab may as well be designed as spanning in only one
direction.
Moments in each direction of span are generally calculated using tabulated
coefÞcients. Areas of reinforcement to resist the moments are determined independently
for each direction of span. The slab is reinforced with bars in both directions parallel to
the spans with the steel for the shorter span placed furthest from the neutral axis to give
it the greater effective depth.
232 Reinforced Concrete Design
Beam A
Figure 8.6
Loads carried by supporting
beams Load on beam A o
45
Beam D
Beam C
Load on Load on
beam C beam D
Load on beam B
Beam B
The span–effective depth ratios are based on the shorter span and the percentage of
reinforcement in that direction.
With a uniformly distributed load the loads on the supporting beams may generally
be apportioned as shown in Þgure 8.6.
8.5.1 Simply supported slab spanning in two directions
A slab simply supported on its four sides will deßect about both axes under load and the
corners will tend to lift and curl up from the supports, causing torsional moments. When
no provision has been made to prevent this lifting or to resist the torsion then the
moment coefÞcients of table 8.4 may be used and the maximum moments are given by
Msx ¼ asx nlx2 in direction of span lx
and
Msy ¼ asy nlx2 in direction of span ly
where
Msx and Msy are the moments at mid-span on strips of unit width with spans lx and ly
respectively
n ¼ ð1:35gk þ 1:5qk Þ, that is the total ultimate load per unit area
ly ¼ the length of the longer side
lx ¼ the length of the shorter side
asx and asy are the moment coefficients from table 8.4.
The area of reinforcement in directions lx and ly respectively are
Msx
Asx ¼ per metre width
0:87fyk z
and
Msy
Asy ¼ per metre width
0:87fyk z
The slab should be reinforced uniformly across the full width, in each direction.
Table 8.4 Bending-moment coefficients for slabs spanning in two directions at right
angles, simply supported on four sides
ly =lx 1.0 1.1 1.2 1.3 1.4 1.5 1.75 2.0
asx 0.062 0.074 0.084 0.093 0.099 0.104 0.113 0.118
asy 0.062 0.061 0.059 0.055 0.051 0.046 0.037 0.029
Design of reinforced concrete slabs 233
The effective depth d used in calculating Asy should be less than that for Asx because
of the different depths of the two layers of reinforcement.
Established practice suggests that at least 40 per cent of the mid-span reinforcement
should extend to the supports and the remaining 60 per cent should extend to within
0:1lx or 0:1ly of the appropriate support.
It should be noted that the above method is not specially mentioned in EC2; however,
as the method was deemed acceptable in BS8110, its continued use should be an
acceptable method of analysing this type of slab.
EXA M PLE 8. 5
Design the reinforcement for a simply supported slab
The slab is 220 mm thick and spans in two directions. The effective span in each
direction is 4.5 m and 6.3 m and the slab supports a variable load of 10 kN/m2. The
characteristic material strengths are fck ¼ 25 N/mm2 and fyk ¼ 500 N/mm2 .
ly =lx ¼ 6:3=4:5 ¼ 1:4
From table 8.4, asx ¼ 0:099 and asy ¼ 0:051.
3
Self-weight of slab ¼ 220 25 10 ¼ 5:5 kN/m2
ultimate load ¼ 1:35gk þ 1:5qk
¼ 1:35 5:5 þ 1:5 10:0 ¼ 22:43 kN/m2
Bending – short span
With class XC-1 exposure conditions take d ¼ 185 mm.
Msx ¼ asx nlx2 ¼ 0:099 22:43 4:52
¼ 45:0 kN m
Msx 45:0 106
¼ ¼ 0:053
bd 2 fck 1000 1852 25
From the lever-arm curve, Þgure 4.5, la ¼ 0:95. Therefore
lever-arm z ¼ 0:95 185 ¼ 176 mm
and
Msx 45:0 106
As ¼ ¼
0:87fyk z 0:87 500 176
¼ 588 mm2 /m
Provide H12 at 175 mm centres, As ¼ 646 mm2 /m.
Span–effective depth ratio
100As; req 100 588
¼ ¼ ¼ 0:318
bd 1000 185
From Þgure 6.3, this corresponds to a basic span–effective depth ratio of 28.0:
span 4500
actual ¼ ¼ 24:3
effective depth 185
Thus d ¼ 185 mm is adequate.
234 Reinforced Concrete Design
Bending – long span
Msy ¼ asy nlx2
¼ 0:051 22:43 4:52
¼ 23:16 kN m
Since the reinforcement for this span will have a reduced effective depth, take
z ¼ 176 12 ¼ 164 mm. Therefore
Msy
As ¼
0:87fyk z
23:16 106
¼
0:87 500 164
¼ 325 mm2 /m
Provide H10 at 200 mm centres, As ¼ 393 mm2 /m
100As 100 393
¼
bd 1000 164
¼ 0:24
which is greater than 0.13, the minimum for transverse steel, with class C25/30 concrete.
The arrangement of the reinforcement is shown in Þgure 8.7.
H10 – 200
Figure 8.7
Simply supported slab
spanning in two directions
H12 – 175
4.5m
8.5.2 Restrained slab spanning in two directions
When the slabs have Þxity at the supports and reinforcement is added to resist torsion
and to prevent the corners of the slab from lifting then the maximum moments per unit
width are given by
Msx ¼ sx nlx2 in direction of span lx
and
Msy ¼ sy nlx2 in direction of span ly
where sx and sy are the moment coefÞcients given in table 8.5, based on previous
experience, for the speciÞed end conditions, and n ¼ ð1:35gk þ 1:5qk Þ, the total
ultimate load per unit area.
The slab is divided into middle and edge strips as shown in Þgure 8.8 and
reinforcement is required in the middle strips to resist Msx and Msy . The arrangement
this reinforcement should take is illustrated in Þgure 8.2. In the edge strips only nominal
reinforcement is necessary, such that As =bd ¼ 0:26fctm =fyk 0:0013 for high yield
steel.
Design of reinforced concrete slabs 235
Table 8.5 Bending moment coefficients for two-way spanning rectangular slabs supported by beams
Short span coefficients for values of ly =lx
Long-span coefficients for
Type of panel and moments considered 1.0 1.25 1.5 1.75 2.0 all values of ly =lx
Interior panels
Negative moment at continuous edge 0.031 0.044 0.053 0.059 0.063 0.032
Positive moment at midspan 0.024 0.034 0.040 0.044 0.048 0.024
One short edge discontinuous
Negative moment at continuous edge 0.039 0.050 0.058 0.063 0.067 0.037
Positive moment at midspan 0.029 0.038 0.043 0.047 0.050 0.028
One long edge discontinuous
Negative moment at continuous edge 0.039 0.059 0.073 0.083 0.089 0.037
Positive moment at midspan 0.030 0.045 0.055 0.062 0.067 0.028
Two adjacent edges discontinuous
Negative moment at continuous edge 0.047 0.066 0.078 0.087 0.093 0.045
Positive moment at midspan 0.036 0.049 0.059 0.065 0.070 0.034
ly ly
lx Figure 8.8
8 Division of slab into middle
and edge strips
Edge strip
Middle strip
Edge strip
Edge strip
lx Middle strip 3lx
4
Edge strip
ly 3ly ly lx
8 4 8 8
(a) For span lx (b) For span ly
In addition, torsion reinforcement is provided at discontinuous corners and it should:
1. consist of top and bottom mats, each having bars in both directions of span;
2. extend from the edges a minimum distance lx =5;
3. at a corner where the slab is discontinuous in both directions have an area of steel in
each of the four layers equal to three-quarters of the area required for the maximum
mid-span moment;
4. at a corner where the slab is discontinuous in one direction only, have an area of
torsion reinforcement only half of that specified in rule 3.
Torsion reinforcement is not, however, necessary at any corner where the slab is
continuous in both directions.
Where ly =lx > 2, the slabs should be designed as spanning in one direction only.
It should be noted that the coefÞcients for both shear and moments can only be used if
class B or C ductility reinforcement is speciÞed and the ratio x=d is limited to 0.25.
236 Reinforced Concrete Design
EX AM PL E 8 .6
Moments in a continuous two-way slab
The panel considered is an edge panel, as shown in Þgure 8.9 and the uniformly
distributed load, n ¼ ð1:35gk þ 1:5qk Þ ¼ 10 kN/m2 .
lx = 5.0m
Figure 8.9
Continuous panel spanning
in two directions
support
a b
Discontinuous
supported edge
ly = 6.25m
support
d c
support
The moment coefÞcients are taken from table 8.5.
ly 6:25
¼ ¼ 1:25
lx 5:0
Positive moments at mid-span
Msx ¼ sx nlx2 ¼ 0:045 10 52
¼ 11:25 kN m in direction lx
Msy ¼ sy nlx2 ¼ 0:028 10 52
¼ 7:0 kN m in direction ly
Negative moments
Support ad, Mx ¼ 0:059 10 52 ¼ 14:75 kN m
Supports ab and dc, My ¼ 0:037 10 52 ¼ 9:25 kN m
The moments calculated are for a metre width of slab.
The design of reinforcement to resist these moments would follow the usual
procedure. Torsion reinforcement, according to rule 4 is required at corners b and c. A
check would also be required on the span–effective depth ratio of the slab.
8.6 Flat slab floors
A ßat slab ßoor is a reinforced concrete slab supported directly by concrete columns
without the use of intermediary beams. The slab may be of constant thickness
throughout or in the area of the column it may be thickened as a drop panel. The column
may also be of constant section or it may be ßared to form a column head or capital.
These various forms of construction are illustrated in Þgure 8.10.
Design of reinforced concrete slabs 237
Figure 8.10
Drop panels and column
heads
Floor without drop Floor with column Floor with drop
panel or column head but no drop panel and column
head panel head
The drop panels are effective in reducing the shearing stresses where the column is
liable to punch through the slab, and they also provide an increased moment of
resistance where the negative moments are greatest.
The ßat slab ßoor has many advantages over the beam and slab ßoor. The simpliÞed
formwork and the reduced storey heights make it more economical. Windows can
extend up to the underside of the slab, and there are no beams to obstruct the light and
the circulation of air. The absence of sharp corners gives greater Þre resistance as there
is less danger of the concrete spalling and exposing the reinforcement. Deßection
requirements will generally govern slab thickness which should not normally be less
than 180 mm for Þre resistance as indicated in table 8.6.
The analysis of a ßat slab structure may be carried out by dividing the structure into a
series of equivalent frames. The moments in these frames may be determined by:
(a) a method of frame analysis such as moment distribution, or the stiffness method on
a computer;
(b) a simplified method using the moment and shear coefficients of table 8.1 subject to
the following requirements:
(i) the lateral stability is not dependent on the slab-column connections;
(ii) the conditions for using table 8.1 described on page 217 are satisfied;
(iii) there are at least three rows of panels of approximately equal span in the
direction being considered;
(iv) the bay size exceeds 30 m2
Table 8.6 Minimum dimensions and axis distance for flat slabs for fire resistance
Standard fire resistance Minimum dimensions (mm)
Slab thickness, hs Axis distance, a
REI 60 180 15
REI 90 200 25
REI 120 200 35
REI 240 200 50
Note:
1. Redistribution of moments not to exceed 15%.
2. For fire resistance R90 and above, 20% of the total top reinforcement in each direction over
intermediate supports should be continuous over the whole span and placed in the column strip.
238 Reinforced Concrete Design
l
Figure 8.11 Position of maximum
Flat slab divided into strips negative moment
Position of maximum
positive moment
cL
Half Half
column Middle column
strip strip strip
Width of half column strip = l/4 with no drops
or = half drop width when drops are used
Interior panels of the ßat slab should be divided as shown in Þgure 8.11 into column
and middle strips. Drop panels should be ignored if their smaller dimension is less than
one-third of the smaller panel dimension lx . If a panel is not square, strip widths in both
directions are based on lx .
Moments determined from a structural analysis or the coefÞcients of table 8.1 are
distributed between the strips as shown in table 8.7 such that the negative and positive
moments resisted by the column and middle strips total 100 per cent in each case.
Reinforcement designed to resist these slab moments may be detailed according to
the simpliÞed rules for slabs, and satisfying normal spacing limits. This should be
spread across the respective strip but, in solid slabs without drops, top steel to resist
negative moments in column strips should have one-half of the area located in the
central quarter-strip width. If the column strip is narrower because of drops, the
moments resisted by the column and middle strips should be adjusted proportionally as
illustrated in example 8.7.
Column moments can be calculated from the analysis of the equivalent frame.
Particular care is needed over the transfer of moments to edge columns. This is to ensure
that there is adequate moment capacity within the slab adjacent to the column since
moments will only be able to be transferred to the edge column by a strip of slab
considerably narrower than the normal internal panel column strip width. As seen in
table 8.7, a limit is placed on the negative moment transferred to an edge column, and
slab reinforcement should be concentrated within width be as deÞned in Þgure 8.12. If
exceeded the moment should be limited to this value and the positive moment increased
to maintain equilibrium.
The reinforcement for a ßat slab should generally be arranged according to the rules
illustrated in Þgure 8.2, but at least 2 bottom bars in each orthogonal direction should
pass through internal columns to enhance robustness.
Table 8.7 Division of moments between strips
Column strip Middle strip
Negative moment at edge column 100% but not more 0
than 0:17be d 2 fck
Negative moment at internal column 60–80% 40–20%
Positive moment in span 50–70% 50–30%
be = width of edge strip.
Design of reinforced concrete slabs 239
cz cz
Figure 8.12
Definition of be
slab edge cy slab edge
cy
y y
slab edge
inner face of column z
be = cz + y be = z + y/2
Note: All slab reinforcement perpendicular to a free edge transferring moment to the column
should be concentrated within the width be
(a) Edge column (b) Corner column
Important features in the design of the slabs are the calculations for punching shear at
the head of the columns and at the change in depth of the slab, if drop panels are used.
The design for shear should follow the procedure described in the previous section on
punching shear except that EC2 requires that the design shear force be increased above
the calculated value by 15 per cent for internal columns, up to 40 per cent for edge
columns and 50 per cent for corner columns, to allow for the effects of moment transfer.
These simpliÞed rules only apply to braced structures where adjacent spans do not differ
by more than 25%.
In considering punching shear, EC2 places additional requirements on the amount
and distribution of reinforcement around column heads to ensure that full punching
shear capacity is developed.
The usual basic span–effective depth ratios may be used but where the greater span
exceeds 8.5 m the basic ratio should be multiplied by 8.5/span. For ßat slabs the span–
effective depth calculation should be based on the longer span.
Reference should be made to codes of practice for further detailed information
describing the requirements for the analysis and design of ßat slabs, including the use of
bent-up bars to provide punching shear resistance.
EXA M PLE 8. 7
Design of a flat slab
The columns are at 6.5 m centres in each direction and the slab supports a variable load
of 5 kN/m2. The characteristic material strengths are fck ¼ 25 N/mm2 for the concrete,
and fyk ¼ 500 N/mm2 for the reinforcement.
It is decided to use a ßoor slab as shown in Þgure 8.13 with 250 mm overall depth of
slab, and drop panels 2.5 m square by 100 mm deep. The column heads are to be made
1.2 m diameter.
Permanent load
Weight of slab ¼ 0:25 25 6:52 ¼ 264:1 kN
Weight of drop ¼ 0:1 25 2:52 ¼ 15:6 kN
Total ¼ 279:7 kN
240 Reinforced Concrete Design
2.5m square drops
Figure 8.13
Flat slab example
h = 250
100
h c =1.2m
6.5m column centres each way
Variable load
Total ¼ 5 6:52 ¼ 211:3 kN
Therefore
ultimate load on the floor, F ¼ 1:35 279:7 þ 1:5 211:3
¼ 695 kN per panel
695
and equivalent distributed load, n ¼ 2 ¼ 16:4 kN/m2
6:5
The effective span,
slab thickness
L ¼ clear span between column heads þ at either end
2
350 3
¼ ð6:5 1:2Þ þ 2 10 ¼ 5:65 m
2
A concrete cover of 25 mm has been allowed, and where there are two equal layers of
reinforcement the effective depth has been taken as the mean depth of the two layers in
calculating the reinforcement areas. (d ¼ 205 mm in span and 305 mm at supports.)
The drop dimension is greater than one-third of the panel dimension, therefore the
column strip is taken as the width of the drop panel (2.5 m).
Bar spacing and size limits
As the slab thickness is greater than 200 mm the spacing limits given in section 8.3(c) do
not apply. Use the spacing or bar size limits from tables 6.7 and 6.9 for the bending
reinforcement. In both cases the steel stress under the quasi-permanent loads can be
estimated from equation 6.1 assuming 20% redistribution (the maximum applicable to
table 8.1 used below) as:
fyk Gk þ 0:3Qk 1 As; req
fs ¼
1:15 ð1:35Gk þ 1:5Qk Þ As; prov
500 ð279:7 þ 0:3 211:3Þ 1
1
1:15 695 0:8
¼ 268 N/mm2
assuming that As; prov ¼ As; req .
Hence from table 6.7 the maximum spacing 165 mm
or from table 6.9 the maximum bar size ¼ 12 mm
