4.4 Solid slabs
4.4.4 Solid slabs spanning in two directions at right angles (uniformly distributed loads) 43
In addition to other methods, the methods given in 4.4.4.1 to 4.4.4.3 may be used for the design of slabs spanning in two directions at right angles and supporting uniformly distributed loads.
4.4.4.1 Simply supported slabs
When simply supported rectangular slabs do not have adequate provision to resist torsion at the corners and to prevent the corners from lifting, the maximum moments per unit width are given by the following equations:
Msx= sxnl2 x
Msy= synl x2
where
Msx, Msy are the maximum bending moments at mid-span on strips of unit width spanning lx and ly, respectively;
n is the total ultimate load per unit area (1,2 gn + 1,6 qn);
lx is the length of shorter side;
ly is the length of larger side; and
αsx, αsy are the bending moment coefficients given in table 14.
Extend to the supports at least 50 % of the tension reinforcement provided at mid-span. Extend the remaining part of the reinforcement to within 0,1lx or 0,1ly of the support, as appropriate.
Table 14 - Bending moment coefficients for slabs spanning in two directions at right angles, simply supported on four sides
1 2 3
Both in continuous and in discontinuous slabs where the corners are prevented from lifting and provision for torsion is made, the maximum moments per unit width are given by the following equations:
Msx = βsxnl2x (3)
Msy = βsynl2x (4)
where
Msx, Msy are the maximum bending moments at mid-span on strips of unit width spanning lx and ly, respectively;
n is the total ultimate load per unit area (1,2 gn + 1,6 qn);
lx is the length of shorter side;
ly is the length of larger side; and
βsx, βsy are the bending moment coefficients given in table 15.
Table 15 - Bending moment coefficients for rectangular panels supported on four sides with provision for torsional reinforcement
at the corners Positive moment at mid-span . . . 0,031
0,024 0,037 Positive moment at mid-span . . .
0,039 Positive moment at mid-span . . .
0,039 Positive moment at mid-span . . .
0,047 Positive moment at mid-span . . .
0,046 Positive moment at mid-span . . .
Positive moment at mid-span . . .
0,057 Positive moment at mid-span . . .
Positive moment at mid-span . . . 0,055 0,065 0,074 0,081 0,087 0,092 0,103 0,111 0,056
Where these equations are used, the conditions given below apply.
4.4.4.2.1 In the case of continuous slabs
The nominal self-weight and imposed loads on adjacent slabs should be approximately the same as those on the slab under consideration, and the spans of all adjacent slabs should be approximately the same in each of the two directions of the lines of the supports. (See also 4.4.4.2.2.)
4.4.4.2.2 In the case of continuous and discontinuous slabs
Regard slabs as divided in each direction into middle strips and edge strips as shown in figure 9, the middle strip being three-quarters of the width and each edge strip one-eighth of the width;
The maximum moments calculated as in 4.4.4.2 apply to the middle strips only and no redistribution is permitted.
Figure 9 — Division of slab into middle and edge strips
a) tension reinforcement at mid-span: extend at least 50 % of the tension reinforcement provided at mid-span in the middle strip in the lower part of the slab to within 0,15l of the continuous edge axis, and to within 50 mm of the discontinuous edge axis; extend the remaining part of the reinforcement to within 0,25l of a continuous edge axis, and to within 0,15l of the discontinuous edge axis;
b) tension reinforcement over the continuous edges: extend at least 50 % of the tension reinforcement provided in the upper part of a middle strip to a distance 0,3l from the face of the support; extend the remaining part of the reinforcement to a distance of 0,15l from the face of the support;
c) tension reinforcement over the discontinuous edge: at a discontinuous edge, negative moments may arise, depending on the degree of fixity of the edge of the slab; in general, tension reinforcement equal to 50 % of that provided at mid-span extending 0,1l into the span (from the face of the support) will be sufficient;
d) tension reinforcement in an edge strip, parallel to the edge: the reinforcement need not exceed the minimum given in 4.11.4 and the minimum given in the rules for torsional reinforcement given in (e), (f) and (g) below;
e) torsional reinforcement at any corner where the slab is simply supported on both edges meeting at that corner: the reinforcement should comprise top and bottom reinforcement, each with layers of bars placed parallel to the sides of the slab and extending from the external faces of the edges a minimum distance of one-fifth of the shorter span; the area of reinforcement in each of these four layers should be three-quarters of the area required for the maximum mid-span moment in the slab;
f) torsional reinforcement at any corner contained by edges over only one of which the slab is continuous: reinforcement equal to half of that described in (e) above should be provided;
g) torsional reinforcement need not be provided at any corner contained by edges over both of which the slab is continuous: where ly /lx exceeds 3, so design slabs as to span one way only.
4.4.4.2.3 In the case of a restrained slab with unequal conditions at adjacent panels
If the support moments for adjacent panels (calculated using table 15) differ significantly, they may be adjusted as follows:
a) calculate the sum of the moments at mid-span and supports (ignoring signs);
b) treat the values from table 15 as fixed end moments;
c) distribute these fixed end moments across the supports according to the relative stiffness of adjacent spans, giving new support moments;
d) adjust the mid-span moment; this should be such that when it is added to the support moments as in (c) above (ignoring signs), the total should equal that obtained in (a) above;
if, for a given panel, the resulting support moments now significantly exceed the values given by equations (3) and (4), the tension steel over the supports will need to be extended beyond the provisions of 4.11.7.3. The procedure is as follows:
1) the span moment is taken as parabolic between supports; its maximum value is as found in (d) above;
2) the points of contraflexure of the new support moments (as in (c) above) and the span moment (as in (1) above) are determined;
3) at each end, half the support tension steel is extended to at least an effective depth or 12 bar diameters beyond the nearest point of contraflexure; and
4) at each end, the full area of the support tension steel is extended to half the distance obtained in (3) above.
4.4.4.3 Loads on supporting beams
The design loads on beams supporting solid slabs spanning in two directions at right angles and supporting uniformly distributed loads may be assumed to be in accordance with figure 10. If the edges of two slabs having the same support meet at a corner, the dividing angle is 45°. If a fully restrained edge meets a freely supported edge, the dividing angle on the restrained side is 60°. With partial restraint, the angles may be assumed to lie between 45° and 60° (see figure 10(b)).
Figure 10 — Apportionment of load for determining the bearing reactions 4.4.5 Shear resistance of solid slabs
4.4.5.1 Shear stresses in solid slabs
The design shear stress v at any cross-section in a solid slab should be compared with the allowable shear stress vc and in no case should it exceed the lesser of 0,75 fcu or 4,75 MPa, whatever reinforcement is provided.
Calculate v from:
(5) v ' V
bd where
v is the design shear stress;
V is the shear force due to design maximum loads;
b is the width of slab under consideration (usually 1 000 mm); and d is the effective depth;
and the allowable stress vc is the maximum design shear stress in concrete without shear reinforcement (obtainable from 4.3.4.1).
When the design shear stress v is less than the allowable shear stress vc, no shear reinforcement is needed.
When v exceeds vc, shear reinforcement should be provided in accordance with the appropriate rules for beams (see 4.3.4). When links are used in slabs less than 200 mm thick, the partial loss of efficiency of the links should be taken into consideration unless structural steel shear heads are provided that have been designed in accordance with specialist literature. It may be assumed that every 10 mm reduction in the slab thickness reduces the links' efficiency by 10 %.
The enhancement in design shear strength close to supports (as described in 4.3.4.2) may also be applied to solid slabs.
4.4.5.2 Shear stresses in solid slabs under concentrated load
4.4.5.2.1 The following terms specific to perimeters are used in this subclause:
a) perimeter: a boundary of the smallest rectangle (or square) that can be drawn around a loaded area and that nowhere comes closer to the edges of the loaded area than some specified distance lp (a multiple of 0,75d) (see figure 11).
NOTE - See 4.4.5.2.8 for loading close to a free edge.
b) failure zone: an area of slab bounded by perimeters 1,5d apart (see figure 12);
c) effective length of a perimeter: the length of the perimeter reduced, where appropriate, for the effects of openings or external edges;
d) effective depth d: the average effective depth for all effective reinforcement passing through a perimeter; and
e) effective steel area: the total area of all tension reinforcement that passes through a zone and that extends at least one effective depth (see above) or 12 times the bar size beyond the zone on either side.
NOTE - The reinforcement percentage used to calculate the design ultimate shear stress vc is given by:
100 x effective steel area ud
where
u is the outer perimeter of zone concerned; and d is the effective depth (as defined above).
Figure 11 — Definition of a shear perimeter for typical cases
4.4.5.2.2 The maximum design shear stress vmax resulting from the concentrated load and calculated as below should not exceed the lesser of 0,75 fcu or 4,75 MPa.
(6) vmax V
uo d where
V is the design maximum value of concentrated load;
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uo is the effective length of perimeter that touches a loaded area; and
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d is the effective depth of slab.
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The maximum shear capacity may also be limited by the provisions of 4.4.5.2.6. Amdt 1, Apr. 1994
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4.4.5.2.3 The shear capacity of punching shear zones is checked first on a perimeter 1,5 d from the
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face of the loaded area. If the calculated shear stress does not exceed vc, then no further checks are
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needed. Amdt 1, Apr. 1994
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If shear reinforcement is required, then it should be provided on at least two perimeters within the zone indicated in figure 12. The first perimeter of reinforcement should be located at approximately 0,5d from the face of the loaded area and should contain not less than 40 % of the calculated area of reinforcement.
The spacing of perimeters of reinforcement should not exceed 0,75d and the spacing of the shear reinforcement around any perimeter should not exceed 1,5d. The shear reinforcement should be anchored round at least one layer of tension reinforcement. The shear stress should then be checked on successive perimeters at 0,75d intervals until a perimeter is reached which does not require shear reinforcement.
In the provision of reinforcement for the shear calculated on the second and subsequent perimeters, the reinforcement provided for the shear on previous perimeters and that lies within the zone shown in figure 12 should be taken into account.
4.4.5.2.4 The nominal design shear stress v, appropriate to a particular perimeter, is calculated from:
(7) | v ' V
ud
Amdt 1, Apr. 1994
where
V, d are as in equation (5); and
u is the effective length of the outer perimeter of the zone.
4.4.5.2.5 No shear reinforcement is required when the stress v is less than vc as calculated in 4.3.4.1. | The value of 100 As/bvd to be used in 4.3.4.1 may be taken as the average for the two directions. |
Amdt 1, Apr. 1994
In the case of zone 1, As in each direction should include all the tension reinforcement within a strip of width bv equal to the width of the loaded area plus three times the effective depth of slab on either
side of the loaded area. Amdt 1, Apr. 1994 |
The enhancement of vc permitted in 4.3.4.2 should not be applied to the shear strength of perimeters at a distance of 1,5d or more from the face of the loaded area. Where it is desired to check perimeters closer to the loaded area than 1,5d, vc may be increased by a factor 1,5 d/av (up to 4 MPa), where av
is the distance from the edge of the loaded area to the perimeter considered.
4.4.5.2.6 The use of shear reinforcement other than links is not covered specifically by this code and |
should be justified separately. Amdt 1,Apr. 1994 |
In slabs over 200 mm, if vc <v <2vc, shear reinforcement may be provided in accordance with equation 7(a) or 7(b), as relevant:
For cases where vc <v <1,6vc, shear reinforcement should be provided in accordance with
ASV > (v & vc)ud (7(a))
0,87fyv
For cases where 1,6vc <v <2vc, shear reinforcement should be provided in accordance with
ASV > 5( 0,7v & vc) ud (7(b))
0,87f
Equations 7(a) and 7(b) should not be applied where the shear stress v exceeds 2vc.
Where v>2vc and a reinforcing system is provided to increase the shear resistance, justification should be provided to demonstrate the validity of the design.
In the above equations:
Asv is the area of shear reinforcement;
fyv is the characteristic strength of shear reinforcement (but not exceeding 450 MPa);
u is the effective length of the outer perimeter of the zone;
d is the effective depth of slab; and v - vc > 0,4 MPa.
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NOTE – d is the average effective depth for all effective reinforcement passing through a perimeter.
Figure 12 — Punching shear zones
4.4.5.2.7 When openings in slabs and footings (see figure 13) are located at a distance of less than six times the thickness of the slab from the edge of a concentrated load or reaction, then that part of the periphery of the critical section that is enclosed by radial projections of the openings to the centroid of the loaded area is to be considered ineffective.
Figure 13 — Openings in slabs
Where an opening is adjacent to the loaded area and its greatest width is less than the lesser of one-quarter of the loaded area side or one-half of the slab depth, its presence may be ignored.
4.4.5.2.8 Where a concentrated load is located close to a free edge, the effective length of a perimeter should be taken as the lesser of the two illustrated in figure 14. The same principle may be adopted for corner columns.
Figure 14 — Shear perimeters with loads close to free edge