Deflection and crack control
3.5 Maximum span-to-depth ratio – minimum thickness
In the design of a reinforced concrete beam or slab, the designer is first confronted with the problem of selecting a suitable depth or thickness of the member. In the case of floor slabs, a reasonable first estimate is desirable since, in many cases, the self-weight of the slab is a large proportion of the total service load. Strength considerations alone may result in a slab thickness that leads to excessive deflection at service loads. The relatively low tensile reinforcement quantities commonly used in slabs (Ast/bd is
typically in the range 0.0025–0.006) are evidence of the importance of serviceability in the selection of slab thickness.
Many codes of practice specify a minimum thickness, h, or a maximum span-to- effective depth ratio,/d. The implication is that deflection will be acceptable if the beam or slab thickness is greater than the minimum value, and thus the deflection need not be calculated.
For example, for normal weight concrete members containing steel reinforcement with a yield stress of fy= 500 MPa and for members that are not supporting or
attached to partitions or other construction likely to be damaged by large deflection, ACI-318-08 (Ref. 4) specifies that deflections need not be calculated if the thickness of a beam or one-way slab is greater than the minimum thickness given in Table 3.3
Table 3.2 Recommended maximum final design crack width, w∗
Environment Design requirement Maximum final crack
width, w* (mm) Sheltered environment
(where crack widths will not adversely affect durability)
Aesthetic requirement
• where cracking could adversely affect the appearance of the structure
• close in buildings • distant in buildings • where cracking will not be
visible and aesthetics is not important.
0.3 0.5 0.7
Environment Durability requirement
• where wide cracks could lead to corrosion of reinforcement
0.3
Aggressive environment
Durability requirement
• where wide cracks could lead to corrosion of reinforcement
0.30 (when c∗≥ 50 mm) 0.25 (otherwise) ∗c is the concrete cover to the nearest steel reinforcement.
Table 3.3 Minimum thickness for non-prestressed beams or one-way slabs – fy= 500 MPa (Ref. 4)
Minimum thickness, h
Simply- One end Both ends Cantilever
supported continuous continuous
Solid one-way slab /18 /21.5 /25 /9
Beam or ribbed slab /14.5 /16.5 /19 /7
is the span measured centre to centre of supports or the clear projection of a cantilever.
or if the thickness of a two-way flat slab or flat plate is greater than the value given in Table 3.4. This deemed-to-comply approach is attractive because of its simplicity and, if it always led to appropriately proportioned and serviceable slabs, it would be ideal for use in routine design. However, in some situations, the use of the minimum thicknesses in Tables 3.3 and 3.4 leads to slabs that are far thicker than they need to be. In other situations, some heavily loaded slabs with the minimum thicknesses specified in Tables 3.3 and 3.4 deflect excessively. Indeed, a single value for minimum thickness that does not account for the load level, the steel quantities and location, the occupancy of the structure (and therefore the desired deflection limit), the load duration, the quality of the concrete and the environment cannot possibly ensure deflection control in every situation, unless, of course, it is grossly conservative in many situations and therefore entirely unsuitable for use in structural design.
Based on the work of Rangan (Ref. 22) and Gilbert (Ref. 23), the Australian Standard AS3600-2009 (Ref. 7) specifies a maximum span to effective depth ratio
Table 3.4 Minimum thickness of two-way slabs without interior beams (flat slabs or flat plates) – fy= 500 MPa (Ref. 4)
Minimum thickness, h
Without drop panels With drop panels
Exterior panels Interior panels Exterior panels Interior panels
Without edge With edge Without edge With edge
beams beams beams beams
n/28 n/31 n/31 n/31 n/34 n/34
nis the clear span measured face to face of supports in long span direction.
for slabs given by:
ef/d = k3k4 (/ef)1000Ec Fd.ef 1/3 (3.1)
whereef is the lesser of the clear span plus slab thickness and the centre-to-centre
distance between supports; d is the effective depth from the compressive surface of the slab to the centroid of the tensile reinforcement; is the deflection limit selected in design (either total or incremental deflection); Ecis the elastic modulus of the concrete
(in MPa); Fd.ef is the effective design service load (in kPa) and is equal to 3g+q1+2q2 for total deflection and 2g+ q1+ 2q2 for incremental deflection; g is the dead load (in kPa); q1is the expected short-term live load (in kPa); q2is the sustained part of the live load (in kPa); and k3is a slab system factor given by:
• k3= 1.0 for one-way slabs and two-way edge-supported slabs; • k3= 0.95 for a two-way flat slab without drop panels; and • k3= 1.05 for a two-way flat slab with drop panels.
The term k4is a factor that depends on the continuity at the supports of the slab and, for a one-way slab or two-way flat slab, k4 equals 1.4 for a simply-supported span, l.75 for the end span of a continuous slab and 2.1 for an interior span of a continuous slab. For a two-way edge-supported slab, k4is given in Table 3.5.
Alternative expressions for the limiting span-to-depth ratios are specified in other codes. For example, according to Eurocode 2 (Ref. 6), if the span-to-depth ratio of a member does not exceed the following limiting values, deflections need not be calculated for members where the maximum total deflection is not to exceed span/250 and where the incremental deflection is not to exceed span/500:
d = K 11+ 1.5 fcρo ρ + 3.2 fc ρo ρ − 1 3/2 ifρ ≤ ρo (3.2a) d = K 11+ 1.5 fc ρo ρ − ρ+ 1 12 fc ρ ρo ifρ > ρo (3.2b)
Table 3.5 Continuity factor k4for a rectangular slab panel supported on four sides – (Ref. 7)
Edge condition Continuity factor, k4
Ratio of long to short span (y/x)
1.0 1.25 1.5 2.0
Four edges continuous 3.60 3.10 2.80 2.50
One short edge discontinuous 3.40 2.90 2.70 2.40
One long edge discontinuous 3.40 2.65 2.40 2.10
Two short edges discontinuous 3.20 2.80 2.60 2.40
Two long edges discontinuous 3.20 2.50 2.00 1.60
Two adjacent edges discontinuous 2.95 2.50 2.25 2.00
Three edges discontinuous (one long edge continuous) 2.70 2.30 2.20 1.95 Three edges discontinuous (one short edge continuous) 2.70 2.10 1.90 1.60
Four edges discontinuous 2.25 1.90 1.70 1.50
where K depends on the structural system, with K= 1.0 for simply supported beams and one-way or two-way spanning slabs; K= 1.3 for end spans of continuous beams and one-way or two-way slabs continuous over one long side; K= 1.5 for interior spans of beams and one-way or two-way spanning slabs; K= 1.2 for flat slabs supported on columns without beams (based on longer span); and K= 0.4 for a cantilever; ρois a
reference reinforcement ratio given byρo= 0.001
fc;ρ is the tension reinforcement
ratio at mid-span required to resist the moment due to the design loads (at the support for cantilevers);ρis the compression reinforcement ratio at mid-span (at the support for cantilevers); fcis in MPa.
For flanged sections where the ratio of the flange width to the web width exceeds 3, the values obtained from Eqs 3.2 should be multiplied by 0.8. For beams and slabs, other than flat slabs, supporting partitions likely to be damaged by excessive deflection and with spans exceeding 7 m, the values of/d given by Eqs 3.2 should be multiplied by 7/ef, whereef is the effective span in metres. For flat slabs supporting partitions
likely to be damaged by excessive deflection and where the longer span exceeds 8.5 m, the values of/d given by Eqs 3.2 should be multiplied by 8.5/ef.
An iterative procedure is required to use Eqs 3.2. An initial estimate of the effective depth d must be made in order to calculate the reinforcement ratiosρ and ρrequired to resist the design moment at the critical sections. These reinforcement ratios are in turn required in the calculation of the limiting span-to-depth ratio using Eqs 3.2. Depending on the initial estimate of d, the required reinforcement ratios may need to be recalculated, together with a revised slab depth, to ensure deflection control. Similarly, the use of Eq. 3.1 requires an initial estimate of slab thickness to determine the self-weight and an iteration may be required if the initial estimate is poor. On the other hand, the ACI 318-08 (Ref. 4) minimum thicknesses do not require any iteration and can be used to select an initial member thickness at the beginning of the design.
Designers should be aware that the use of either the ACI 318-08 minimum thicknesses (Tables 3.3 and 3.4) or the AS3600-2009 (Ref. 7) maximum span-to- depth ratio (Eq. 3.1) is inevitably conservative for most practical situations and leads
to beam and slab thicknesses that may be considerably larger than they need to be. In contrast, the Eurocode 2 (Ref. 6) limiting span-to-depth ratios may result in beam or slab thicknesses that are unacceptably small, particularly in situations where the sustained load is a large proportion of the total load, and should not be regarded as a guarantee that deflections will not be excessive.
Example 3.1
The thickness of the end span of a one-way slab is to be estimated for the cases where:
1) the maximum long-term mid-span deflection is not to exceed span/250; and
2) the incremental deflection is not to exceed span/480.
The span is = ef = 5.0 m; the dead load g = 2.0 kPa + self-weight; and the
live load is q= 3.0 kPa. The short-term expected live load is q1= 2.1 kPa and the sustained part of the live load q2= 1.2 kPa. The specified concrete and reinforcement strengths are fc= 25 MPa and fy= 500 MPa; the clear concrete
cover to the longitudinal reinforcement is 20 mm; and the reinforcement bar diameter is db= 12 mm.
ACI318-08
From Table 3.3, if deflections are not to be calculated, the minimum thickness according to ACI318-08 is h = 5000/21.5 = 230 mm, irrespective of the deflection requirements for the slab and irrespective of the loads to be applied to the slab.
AS3600-2009
From Table 2.1, Ec= 26700 MPa. To calculate the maximum effective span-to-
depth ratio using Eq. 3.1, an estimate of self-weight of the slab is required. Take self-weight = 0.23 × 24 = 5.5 kPa and the dead load is therefore g= 2.0 + 5.5 = 7.5 kPa.
(i) If the maximum total deflection is = 5000/250 = 20 mm, then the effective design service load is Fd.ef = 3g + q1+ 2q2= 3 × 7.5 + 2.1 + 2 × 1.2 = 27.0 kPa and, from Eq. 3.1, the maximum span to effective depth ratio is: ef/d = 1.0 × 1.75 (20/5000) × 1000 × 26700 27.0 1/3 = 27.7
The minimum effective depth according to AS 3600-2009 is therefore d= 5000/27.7 = 181 mm and the minimum slab thickness is h = d +db/2+
(ii) If the incremental deflection is = 5000/480 = 10.4 mm, then the effective design service load is Fd.ef = 2g + q1+ 2q2 = 2 × 7.5 + 2.1 + 2 × 1.2 = 19.5 kPa and, from Eq. 3.1, the maximum span to effective depth ratio is:
ef/d = 1.0 × 1.75 (10.4/5000) × 1000 × 26700 19.5 1/3 = 24.8
The minimum effective depth according to AS 3600-2009 is therefore d= 5000/24.8 = 201 mm and the minimum slab thickness is h = d +db/2+
cover= 201 + 6 + 20 = 227 mm.
Eurocode 2
If the design moment at mid-span is w∗2/12, where w∗is the factored design load for the strength limit state, and if after several iterations the initial estimate of d is 150 mm, the reinforcement ratio required to resist the design moment is ρ = 0.0035 and ρ= 0. The reference reinforcement ratio is ρo= 0.001 f
c=
0.005 and the factor K = 1.3. From Eq. 3.2a: d= 1.3 11+ 1.5√25 0.005 0.0035+ 3.2 √ 25 0.005 0.0035− 1 3/2 = 34.1
The minimum effective depth according to Eurocode 2 is therefore d = 5000/34.1 = 147 mm and the minimum slab thickness is h = d +db/2+cover =
147+ 6 + 20 = 173 mm.
Clearly, in this example, the minimum slab thickness obtained using Eq. 3.2a from Eurocode 2 is significantly smaller than the value obtained using either the ACI 318-08 minimum thickness provisions or the maximum span to depth ratio specified in AS 3600-2009.