ACI STRUCTURAL JOURNAL
TECHNICAL PAPER
Title no. 108-S43
Span/Thickness Limits for Deflection Control by Noel J. Gardner Predicting the deflection serviceability of reinforced concrete members is fraught with uncertainties, which include imperfect knowledge of the limiting serviceability criteria, the material properties, and the load history including including construction loads and the service load. The serviceability criteria can be immediate deflection/curvature or incremental deflection/curvature. Most codes offer two methods for control of deflections. The designer may choose to calculate the deflections and check that these computed deflections are less than specified allowable limits. Alternatively Alternatively,, the codes give specified specified maximum span-depth span-depth ratios for which which service serviceabi abilit lityy can can be assumed assumed to be satisf satisfied ied and deflec deflections tions do not need to be calculated. This paper compares the deemed-tocomply span/thickness limits of ACI 318-08, CSA A23.3-04, BS 8110-97, AS 3600-2009, Eurocode 2 (2004), ACI Committee 435 revisions, and the proposals of numerous other authors. Keywords: code provisions; deflection; serviceability. serviceability.
Table 1—Maximum permissible computed deflection (ACI 318-08 and CSA A23.3-04) Type of member
Deflection to be considered
Deflection limitation
Flat roofs not supporting supporting or attached to nonstructural elements Immediate deflection due likely to be damaged by large to live load L deflections
ln /180
Floors not supporting supporting or attached Immediate deflection due to nonstructural elements likely to to live load L be damaged by large deflections
ln /360
Roof or floor construction supporting or attached to nonstructural elements likely to be damaged by large deflections
ln /480
That part of the total deflection occurring after attachment of nonstructural nonstructural elements (sum of the longterm deflection due to all Roof or floor construction sustained loads and the supporting or attached to nonstructural elements not likely likely immediate deflection due to be damaged by large deflections to any additional live load)
ln /240
Note: ln = clear span.
INTRODUCTION The object of structural design is to achieve acceptable probabilities that structures will perform satisfactorily during their intended service life. For safety, the structure must have adequate strength with a low probability of collapse. The required probability against collapse is achieved by increasing the specified loads by appropriate load factors and reducing the member strengths by strength or behavior reduction factors. Although safety is the most important limit state, it is not sufficient without satisfying the requirements of serviceability. Service load deflections/ curvatures may be excessive, or long-term deflections/ curvatures due to sustained loads may cause damage to partitions, visual discomfort, and/or perception. With the increasing use of higher strength concretes and reinforcing steels, as well as more efficient design procedures, there is a tendency toward designing shallower section members in reinforced concrete structures with attendant reductions in stiffness and, hence, larger deflections. The recent (2005) reductions in the ACI load factors have decreased member sizes, increasing the service load/design ultimate load ratio and the possibility of deflection serviceability problems. Most codes offer two methods for control of deflections. The designer may choose to calculate the deflections and check that these computed deflections are less than specified allowable limits. Calculating the immediate deflections of reinforced concrete members is difficult due to the concrete cracking in the tension zones due to early-age construction loads or being under service load. Calculating the additional deflections due to shrinkage, creep, and the consequent redistribution of stress is extremely difficult. Alternatively, the codes give specified maximum span-depth ratios for which serviceability can be assumed to be satisfied and deflections do not need to be calculated.
ACI Structural Journal/July-August 2011
RESEARCH SIGNIFICANCE Codes give specified maximum span/thickness or span/ effective depth ratios for which serviceability can be assumed to be satisfied and deflections do not need to be calculated. The use of higher strength reinforcing steel, more efficient calculation methods, faster construction schedules, and changes in load factors increase the possibility of deflection serviceability problems and warrant a review of current code provisions.
CODE REQUIREMENTS FOR DEFLECTION CONTROL ACI 318-08—Building Code Requirements for Reinforced Concrete; CSA A23.3-04—Design of Concrete Structure for Buildings
The American Code, ACI 318-08,1 and the Canadian Code, CSA A23.3-04,2 are the commonly used design codes for reinforced concrete structures in North America. For beams, their provisions are effectively identical to those in ACI 318-71. The deflection limits are given in Table 1. The minimum thicknesses of beams and one-way slabs not supporting, or attached to, partitions likely to be damaged by large deflections required by both codes are reproduced in Table 2. No guidance is given for beams and slabs supporting or attached to partitions likely to be damaged by deflections. Table 3 is an extended version of Table 2 recommended recom mended by ACI Committee 435,3 which distinguishes between members that support, or are attached to, nonstructural elements likely to be damaged by large deflections and those that do not. Grossman 4 noted that the minimum member ACI Structural Journal, V. 108, No. 4, July-August 2011. MS No. S-2009-389.R1 received May 3, 2010, and reviewed under Institute publication policies. Copyright © 2011, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the May-June 2012 ACI Structu Structural ral Journal Journal if the discussion is received by January 1, 2012.
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( 0.8 + f y / A ) h mi n ≥ l n --------------------------- for α > 2.0 36 + 9 β
Noel J. Gardner , FACI, is a member of ACI Committees 209, Creep and Shrinkage of Concrete; 231, Properties of Concrete at Early Ages; 347, Formwork for Concrete; and 435, Deflection of Concrete Building Structures. His research interests are early-age member behavior, shrinkage, creep, deflection serviceability, and formwork pressures.
where A = 1400 (SI units); A = 200,000 (U.S. customary units); hmin is the slab thickness; ln is the longer clear span; f y is the yield strength of tensile flexural reinforcement (MPa for SI and psi for U.S. units); α is the ratio of flexural stiffness of beam to flexural stiffness of slab; α fm is the average value of α; and β is the ratio of long side to short side. At discontinuous edges, an edge beam with a stiffness α f > 0.8 must be provided or thickness of the panel with a discontinuous edge must be increased by 10%. For flat slabs with drop panels, meeting code-specified minimum thickness and dimensions, the slab thickness beyond the drop panel may be reduced by 10%. CSA A23.3-042 adopted the more conservative provisions proposed by Thompson and Scanlon 5 (for flat slabs without edge beams, use αm = 0).
depths provided in Table 2 (ACI 318-08 Table 9-5(a)), to eliminate the need to calculate deflections, do not correlate with the requirements of Table 1 (ACI 318-08 Table 9-5(b)) of the Code. It can be noted that Table 2 does not take account of several parameters that play important roles in the long-term behavior of reinforced concrete members, that is, the effect of compression steel. Consideration should also be taken for the effect of concrete compressive strength and the magnitude of the service load relative to the ultimate load (a proxy to the extent of tension cracking of the concrete). For slabs, the provisions of the two codes are slightly different. ACI 318-08 requirements for slabs without interior beams or slabs with beams spanning between supports on all four sides with α fm < 0.2, the minimum thickness is given in Table 4. For slabs with beams spanning between the supports on all sides, the minimum thickness is
( 0.8 + f y / A ) h mi n ≥ l n ---------------------------------------------36 + 5 β ( α fm – 0.2 )
for 0.2 < α < 2.0
( 0.6 + f y / B ) h mi n ≥ l n ---------------------------30 + 4 βα m
At discontinuous edges, an edge beam with a stiffness ratio α f > 0.8 must be provided or thickness of the panel with a discontinuous edge must be increased by 10%. For slabs with drop panels, the minimum thickness is given by Eq. (4), where hs is the slab thickness, hd is the total depth of the drop panel, and x d is the distance from the face of the column to the edge of the drop panel.
One end Both ends continuous continuous Cantilever
Members not supporting or attached to partitions or other construction likely to be damaged by large deflection
Member
(3)
(1)
Table 2—Minimum thickness of non-prestressed beams and one-way slabs unless deflections are computed (ACI 318-08 and CSA A23.3-04) Simply supported
(2)
Solid one-way slabs
ln /20
ln /24
ln /28
ln /10
Beams or ribbed one-way slabs
ln /16
ln /18.5
ln /21
ln /8
( 0.6 + f y / B ) 2 x d h mi n ≥ l n ---------------------------– -------- ( h d – h s ) ln 30
(4)
where B = 1000 (SI units); B = 145,000 (U.S. customary units); and f y is yield strength of tensile flexural reinforcement (MPa for SI and psi for U.S. units).
Note: For f y other than 60,000 psi (414 MPa), the values shall be multiplied by 0.4 + f y /100,000 psi units (0.4 + f y /690 SI units).
Table 3—Minimum thickness of beams and one-way slabs used in roof and floor construction (ACI Committee 435 1978) Member
Members not supporting, or not attached to, nonstructural elements likely to be damaged by large deflections
Members supporting, or attached to, nonstructural elements likely to be damaged by large deflections
Simply supported
One end continuous
Both ends continuous
Cantilever
Simply supported
One end continuous
Both ends continuous
Cantilever
Roof slab
ln /22
ln /28
ln /35
ln /9
ln /14
ln /18
ln /22
ln /5.5
Floor slab and roof beam or ribbed roof slab
ln /18
ln /23
ln /28
ln /7
ln /12
ln /15
ln /19
ln /5
Floor beam or ribbed floor slab
ln /14
ln /18
ln /21
ln /5.5
ln /10
ln /13
ln /16
ln /4
Table 4—Minimum thickness of slabs without interior beams unless deflections are computed (ACI 318-08) Without drop panels Exterior panels
With drop panels Interior panels
Without edge beams With edge beams*
Exterior panels
Interior panels
Without edge beams With edge beams*
f y, MPa
f y, psi
280
40,000
ln /33
ln /36
ln /36
ln /36
ln /40
ln /40
420
60,000
ln /30
ln /33
ln /33
ln /33
ln /36
ln /36
520
75,000
ln /28
ln /31
ln /31
ln /31
ln /34
ln /34
*
Slabs with beams along exterior edges. The value of α f for the edge beam shall not be less than 0.8.
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ACI Structural Journal/July-August 2011
Table 6—Modification factor for tension reinforcement (Table 3.10, BS 8110-1997)
Table 5—Basic span/effective depth ratios for beams (Table 3.9, BS 8110-97) Basic span/effective depth ratio Support conditions
Rectangular sections
Flanged beams bw / b < 0.3
Cantilever
7
5.6
Simply supported
20
16.0
Continuous
26
20.8
At discontinuous edges, an edge beam with a stiffness ratio α f > 0.8 must be provided or thickness of the panel with a discontinuous edge must be increased by 10%. The span/thickness provisions of ACI 318-08 and CSA A23.3-04 do not address the sensitivity of slab deflections to early-age construction loads, rate of construction, or concrete strength.
BS 8110-19976—Code of Practice for Design and Construction of Concrete Structures
Eurocode 2-048
Eurocode 28 requires the calculated deflection of a beam or slab subjected to quasi-permanent loads should not exceed span/250. A deflection (incremental) limit after construction of span/500 is normally considered an appropriate limit to avoid deflections that could damage adjacent parts of the structure. The limiting span/depth may be estimated using Eq. (5a) and (5b) modified by factors for boundary conditions and type of reinforcement. 3/2
ρ ρ′ l 1 --- ≤ K 11 + 1.5 f ck / A ---------0----- + ------ f ck / A ----ρ – ρ′ 12 ρ0 d
if ρ < ρ0
(5a)
if ρ > ρ0
(5b)
where A = 1 MPa units (145 psi units), f ck 28 is the 28-day characteristic concrete strength, l / d is the limiting span/
ACI Structural Journal/July-August 2011
Nondimensional moment M u / bd 2, MPa (psi) 0.50 0.75 1.00 1.50 2.00 3.00 4.00 5.00 6.00 (72) (109) (145) (218) (290) (435) (580) (725) (870)
MPa
ksi
100
14.5
2.00
2.00
2.00
1.86
1.63
1.36
1.19
1.08
1.01
150
21.8
2.00
2.00
1.98
1.69
1.49
1.26
1.11
1.01
0.94
200
29.0
2.00
1.95
1.76
1.51
1.35
1.14
1.02
0.94
0.88
250
36.3
1.90
1.70
1.55
1.34
1.20
1.04
0.94
0.87
0.82
300
41.8
1.60
1.44
1.33
1.16
1.06
0.93
0.85
0.80
0.76
307
44.5
1.56
1.41
1.30
1.14
1.04
0.93
0.85
0.80
0.76
Table 7—Modification factor for compression reinforcement (Table 3.11, BS 8110-1997) Reinforcement ratio of compression reinforcement 0.15 0.25 0.50 0.75 1.0 100 As′ / bd Factor to be applied
The provisions of British Standard BS 8110-97, 6 the current evolution of the British Code of Practice CP 110-72 and BS 8110-85, were based on the work of Beeby. 7 Between the 1985 standard and the 1997 standard, the steel material partial safety factor γm changed from 1.15 to 1.05. The span/effective depth requirements for rectangular or flanged beams are based on limiting the total deflection to span/250. These span/effective depth ratios should normally ensure that the part of the deflection occurring after construction of finishes and partitions will be limited to span/350 or 20 mm (0.8 in.), whichever is less, for spans up to 10 m (34 ft). The basic ratios are given in Table 5. The basic ratios are modified according to the ratios of tension and compression reinforcement provided and the service load steel stress at the center of the span (or at the support in the case of a cantilever). These factors are listed in Tables 6 and 7. The span/effective depth ratios take account of normal shrinkage (< 750 × 10 –6) and normal creep (creep coefficient < 3). Tables 5 and 6 can also be used for slabs using the reinforcement ratio at midspan. The reinforcement ratio for a two-way slab supported by walls or stiff beams should be based on the shorter span and the reinforcement ratio in that direction and the longer span for flat slabs.
ρ ρ l --- ≤ K 11 + 1.5 f ck / A -----0 + 3.2 f ck / A ----0 – 1 ρ ρ d
Steel service stress
1.5
2.0
3.0 or greater
1.05 1.08 1.14 1.20 1.25 1.33 1.40 1.50
Table 8—Basic ratios of span/effective depth for reinforced concrete members (Table 7.4N, Eurocode 2-04) Structural system Simply supported beam or two-way simply supported slab
K
Steel ratio = Steel ratio = 1.5% 0.5%
1.0
14
20
End span of continuous beam or one-way continuous slab or two-way slab continuous 1.3 over one long side
18
26
Interior span of continuous beam or two-way slab
1.5
20
30
Flat slab (based on longer span)
1.2
17
24
Cantilever
0.4
6
8
effective depth, K is the structural system factor (Table 8), ρ is the midspan tensile steel ratio, ρ′ is the midspan compression steel ratio, and ρo is the reference reinforcement ratio = 0.001( f ck )1/2 (MPa units) [0.001( f ck × 145)1/2 (psi units)]. Equations (5a) and (5b) were derived assuming the midspan steel stress at the serviceability limit state is 310 MPa (44,000 psi). For flanged sections where the ratio of flange breadth to web breadth exceeds 3, the values should be multiplied by 0.8. For beams and slabs, other than flat slabs, with spans exceeding 7 m (23 ft), which support partitions liable to be damaged by excessive deflections, the l / d values should be multiplied by 7/ l (l in meters) or 23/ l (l in feet). For flat slabs, with spans exceeding 8.5 m (28 ft), which support partitions liable to be damaged by excessive deflections, the l / d values should be multiplied by 8.5/ l (l in meters) or 28/ l (l in feet). Table 8 gives the limiting span/effective depth ratios for beams spanning up to 7 m (23 ft) and flat slabs spanning up to 8.5 m (28 ft) derived on the assumption that the steel stress at midspan is 310 N/mm 2 (44 ksi) and the concrete characteristic strength is 30 MPa (4.4 ksi). For two-way slabs, the calculation should be based on the shorter span and on the longer span for flat slabs. The limits for flat slabs correspond to a less severe limitation than a midspan deflection of span/250 relative to the columns.
AS 3600-2009 Australian Standard9—concrete structures The serviceability requirements of the Australian Standard AS 3600-20099 limit the total deflection to span/250 and the
455
incremental deflection to span/500 where a provision is made to minimize the effect of movement; otherwise, span/1000. Limiting, deemed-to-comply, span-depth ratios for beams can be calculated from the following equation l ef f k 1 ( ∆ / L ef f ) b ef E c ------ ≤ ------------------------------------d k 2 F d . ef
1/3
(6)
where ∆ / leff is the total or incremental deflection limit, beff is the effective width, D is the dead load, E c is the modulus of elasticity of concrete, and F d.ef is the effective design load/unit length. a) (1.0 + k cs) D + (ψs + k csψl) L for total deflection b) k cs D + (ψs + k csψl) L for incremental deflection k 1 = I ef / bef d 3; k 2 = deflection constant 5/384, 2.4/384, and 1.5/384 for simply supported, one end continuous, and interior span, respectively; k cs = [2 – 1.2 As′ / As] > 0.8; leff = effective span; L = live load; ψl = 0.25 for offices and domestic occupancy (0.5 to 0.8 for storage); and ψs = 0.5 for offices (1.0 for storage). A similar equation is given for deem-to-comply spandepth ratios for one-way flat slabs and slabs supported on four sides by walls or stiff beams. l ef f ( ∆ / L ef f ) 1000 E c ------ ≤ k 3 k 4 ------------------------------------d F d . ef
1/3
(7)
where D is the dead load, E c is the modulus of elasticity of concrete, and F d.ef is the effective design load/unit area. a) (1.0 + k cs) D + (ψs + k csψl) L for total deflection b) k cs D + (ψs + k csψl) L for incremental deflection k cs = [2 – 1.2 As′ / As] > 0.8; k 3 = 1.0 for a one-way slab; = 0.95 for a two-way flat slab without drop panels; = 1.05 for a two-way flat slab with drop panels; k 4 = deflection constant 1.4 for simply supported slabs, 1.75 for an end span, or 2.1 for an interior span; L = live load; and leff = effective span. For two-way slabs supported by walls or stiff beams, k 3 = 1.0 and k 4 is given in a table as a function of boundary condit ion and panel aspect ratio.
Gardner and Zhang10—beams Using a layered, nonlinear finite element program, Gardner and Zhang10 determined the span thickness requirements to satisfy a specified deflection limit in terms of specified, or characteristic, concrete strength; tension and compression steel ratios; and the ratio of the sustained moment to the moment capacity of the beam. A hybrid method was used to calculate the long-term behavior using a reduced modulus to account for creep; a conventional time-dependent load vector was used for shrinkage. The positive reinforcement was reduced at the theoretical 50% cutoff point. Characteristic concrete strengths of 20, 30, and 40 MPa (2900, 4400, and 5800 psi) were considered. To take advantage of the mean concrete strengths being larger than the characteristic concrete strengths, the mean concrete
456
strengths were determined using f cm ′ = f ck ′ + 8 MPa ( f cm′ = f ck ′ + 1160 psi), implying shrinkage strains and creep coefficients of 700, 660, and 590 × 10–6; and 2.72, 2.51, and 2.37, respectively. The span/thickness ratio requirements for simply supported beams, satisfying the span/500 deflection criterion under a service load/ultimate load ratio of 50%, are given in Table 9. For the same service moment/design ultimate moment ratio, the span-depth ratios for deflection limits other than span/500, the limiting span-depth ratio is simply multiplied by 500/required span deflection ratio, that is, span-depth ratios for a span/250 deflection criterion can be obtained by doubling the values for the span/500 deflection criterion. The immediate deflection limit of span/375 was not found to be critical. Span-depth ratios for continuous beams may be obtained by multiplying the values for simply supported beams, using the positive moment steel ratio, by the following factors: Support condition Simply supported: One end continuous discontinuous end unrestrained: discontinuous end integral with support: Both ends continuous: Cantilever:
Factor 1.0 1.2 1.3 1.4 0.35
The modifying factors were determined assuming curvature is proportional to the moment coefficients given in ACI 318-08, Section 8.3.3, and CSA A23.3-04, Section 9.3.3. The required span/thickness ratio for a specified deflection limit decreases with an increase in tensile steel ratio, an increase in service moment/ultimate moment, and decreases with an increase in compression reinforcement and an increase in concrete strength. Increasing the service moment, as a fraction of the beam section design ultimate moment, reduces the limiting span-depth ratio. As a first approximation, the limiting span-depth ratio is inversely proportional to the cube root of the ratio of the moment levels. Similarly, it can be deduced that using a higher yield strength steel, which will increase the concrete stress for a given service moment/ design ultimate moment ratio, will also result in smaller permissible span-depth ratios.
Scanlon and Choi11—one-way slabs
Scanlon and Choi11 proposed the following equation based on an incremental deflection limit. ln --h
32 E c α b ∆in c ≤ --------------------------------------------------l n K ( λ W s + W L ( va r ) )
1/3
(8)
where b is the width of beam; E c is the modulus of elasticity for concrete; W L(var ) is the variable portion of live load; W s is the sustained load; α is the ratio of I effective to I gross; λ is the ACI 318 long-term deflection multiplier; and K is the deflection coefficient = 5, 2, and 1.4 for simply supported, one end continuous, and both ends continuous, respectively. This expression requires an iterative procedure to determine the minimum thickness.
Bischoff and Scanlon12—beams and one-way slabs
Bischoff and Scanlon 12 derived expressions to determine limiting span/thickness ratios including the effects of reinforcement ratio, shrinkage restraint, construction loads, sustained live load, support conditions, and deflection limits.
ACI Structural Journal/July-August 2011
For rectangular section members, the following expression was given. Bischoff 13 has proposed an alternative formulation to determine I e that can be used in Eq. (9). 0.8 E c ( I e, D + L / I g ) l --- ≤ ------------------------------------------------------2 h K Ω ( φ / α D + L ) ( d / h ) R n
∆al l -------l
(9)
where E c is the modulus of elasticity for concrete; I e,D+L is the effective moment of inertia under full service load; I e,sus is the effective moment of inertia under sustained load; I g is the gross (uncracked) second moment of area; K is the end restraint factor = 1, 0.85, and 0.8 for simply supported, one end continuous, and both ends continuous, respectively; M n is the nominal moment capacity; Rn is the nominal flexural resistance factor M n / bd 2; α D+L is the average load factor; ∆all is the permissible (allowable) deflection; γ is the ratio of sustained load to full service load; λ is the ACI 318 long I e,sus)]; term deflection multiplier; Ω = {1 + γ(λ – 1)( I e,D+L / and φ = 0.9 strength reduction factor. Results from a comparative study showed that lightly reinforced slabs or beams satisfying the ACI minimum thickness requirement may not satisfy the l /240 incremental deflection requirement.
Thompson and Scanlon5—flat slabs
Thompson and Scanlon5 reported the results of a parametric study of the effects of restraint cracking, concrete strength, design live load, construction load, and panel aspect ratio on the deflections of flat slabs. Deflections were calculated using a plate-bending finite-element program with an effective second moment of area to account for the reduced stiffness due to cracking. It was observed that the calculated deflections were sensitive to the assumed value of the modulus of rupture. Thompson and Scanlon 5 used serviceability criteria of incremental deflection less than span/480 and total deflection less than span/240. From their parametric study, Thompson and Scanlon 5 proposed a more conservative minimum thickness requirement, for both interior and edge panels, for the control of deflections of two-way slabs. The live load deflection limit of span/360 was not found to be critical.
structures leads to the imposition of large early-age construction loads,15 typically of the same order of magnitude as the service loads, on the partially cured supporting slabs. Consequently, it is necessary that both the construction load and design load be taken into account during the design phase of reinforced concrete floor slab construction. The appropriate serviceability criterion will depend on the location of the critical location, midpanel, or column line, and can be immediate or incremental deflection, slope, or curvature. Assuming one level of forms and three levels of reshores (1 + 3), the construction load is 1.25 D ( D is the self-weight of the slab). Assuming the formwork weighs 0.1 D and no construction live load, the total unfactored construction load is 1.35 D.15 Slabs were loaded at 3, 4, 7, 14, and 28 days after casting. At 28 days, the construction load was removed, reducing the slab load to self-weight. Assuming the slab is put to service at 28 days, it is subjected to its own self-weight plus some fraction of the live load. The sustained load was chosen to be self-weight plus 50% of the live load plus a superimposed dead load of 0.1 D. A layered finite element program was used to study the effects of age of imposition of construction loading (age of supporting slab when successive slab is cast), span, panel aspect ratio, live load to dead load ratio, and concrete strength on the deflection serviceability of flat slab systems. 15 It was determined that the age of loading (age superimposed slab cast-construction cycle) and span have significant effects on the slab thickness required to satisfy serviceability. The following equation summarizes the slab thicknesses required to satisfy an exterior panel, interior column line incremental (28 to 5000 days of sustained load) deflection of clear span/240. 1.5
k k l 38 h ≥ ----1------2 ---n0.2 --- ---------53.4 t o f cm 28
(10)
where k (β) = (1.20 – 0.20β) > 0.9 and β is the ratio of longer clear span to shorter clear span. Thompson and Scanlon 5 also recommended that the minimum slab thickness could be reduced by 10% for flat slabs with drop panels whose thickness is greater than or equal to 1.25 times the slab thickness, or by 20% if the drop panel thickness is greater than 1.50 times the slab thickness. Thompson and Scanlon 5 did not investigate the effect of age at which the construction load was applied on the calculated deflections.
Ofosu-Asamoah and Gardner14—flat slabs The deflection serviceability of flat slabs is determined by the loads imposed during the construction process (method and rate of construction), taking account of the concrete strength available when the construction loads are imposed, and the expected sustained service load. The shore-reshore procedure used to construct many reinforced concrete flat slab
ACI Structural Journal/July-August 2011
1.4 + 1.7 L / D -------------------------------2.25
0.25
( 1.15 – 0.15 β )
(11)
where f ck 28 is the 28-day characteristic concrete strength, MPa; f cm28 = f ck 28 + 8 = 28-day mean concrete strength, MPa; h is the slab thickness, m; and ln is the longer clear span, m. 1.5
ln h mi n ≥ ----- k ( β ) 30
0.6
k k l 5500 h ≥ ----1------2 ---n--- ----------- f 96.7 t 0.2 cm 28 o
0.6
1.4 + 1.7 L / D -------------------------------2.25
0.25
( 1.15 – 0.15 β )
(A11)
where f c′28 is the 28-day specified concrete strength, psi; f cm28 = f c′28 + 1160 = 28-day mean concrete strength, psi; h is the slab thickness, ft; ln is the longer clear span, ft; β is the ratio of long clear span to short clear span; k 1 = 0.9 for an interior panel and 1.0 for edge and corner panels; k 2 = 0.9 for slabs with drop panels; t o is the age at which the construction load is applied to the slab; L is live load; and D is dead load. The use of the code-recommended minimum drop panel thickness of 1.25 times the slab thickness reduces the slab thickness required to satisfy the serviceability criterion by approximately 18%; hence, the ACI 318-08 recommendation of a 10% reduction in slab thickness is conservative. For interior panels, the thickness given by Eq. (11) can be reduced by 10% as recommended by ACI 318-08.
DISCUSSION The methods of determining limiting span-depth ratios fall into two main categories: modified elastic beam analysis9,11,12 and parametric studies using finite element
457
Table 9—Proposed span/thickness requirements to satisfy span/500 incremental deflection limit* M = 30% M u
M = 50% M u
M = 70% M u
ρ, %
ρ′, %
f ck = 30 MPa (4400 psi)
20 MPa (2900 psi)
30 MPa (4400 psi)
40 MPa (5800 psi)
20 MPa (2900 psi)
30 MPa (4400 psi)
40 MPa (5800 psi)
< 0.5
0
12.7
8.2
9.8
10.9
7.9
9.8
10.3
1.0
0
11.1
8.4
9.9
10.8
6.8
8.5
9.9
1.5
0
10.8
7.7
8.8
10.1
6.4
8.1
9.2
2.0
0
9.7
7.1
8.3
9.3
5.9
7.4
8.3
1.5
0.5
13.9
11.0
11.6
12.7
8.8
10.7
11.5
2.0
0.5
12.6
9.9
10.8
11.6
8.0
9.7
10.3
2.5
0.5
11.8
9.2
10.2
10.8
7.7
8.7
9.5
2.0
1.0
15.0
12.6
13.3
14.2
10.2
11.9
12.7
2.5
1.0
14.2
11.5
12.4
12.6
9.9
10.8
11.3
3.0
1.0
13.7
11.0
11.4
12.3
9.2
10.1
10.6
2.5
1.5
16.6
14.0
15.0
14.6
12.3
13.2
13.5
3.0
1.5
14.7
13.3
13.7
14.0
11.2
12.2
13.0
*
For deflection limit of span/250, multiply values by 2.
analysis.5,6,10,14 All beam analyses use an effective moment of inertia to approximate the extent of cracking. Finite element calculations can be done using an effective moment of inertia or using several layers through the thickness of the member. Long-term deflections can be done using a simple, combined shrinkage and creep multiplier on the calculated immediate deflection(s) or summing separate calculations for the deflections caused by load effects with the deflection due to shrinkage. A single long-term deflection multiplier 9,11,12 to take account of both shrinkage and creep should include load magnitude and reinforcement ratio. All results are consequences of the input data assumptions, namely, magnitude of service loads (assumed applied at 28 days), concrete strength, reinforcement ratio, and, for flat slabs, the age of imposition and magnitude of the construction loads. Obviously the deflections calculated for members designed using thicknesses given by deemed-to-comply provisions should satisfy the code-specified deflection limitations. While the characteristic live load—live load not exceeded 95% of the time—should be used for ultimate limit state calculations, the experimental work of Choi 16 indicated that an average live load of 50% of the extreme (assumed specified/characteristic) live load would be reasonable for serviceability limit state calculations. The percentage is dependent on the load factors and the material/member understrength factors used in the ultimate limit state calculations. BS 8110-976 states that when calculating deflections, the portion of the live load to be considered permanent should be 25 to 30% for office use but at least 75% for storage. AS 3600-20099 suggests that for deflection calculations, the characteristic live load can be multiplied by 0.6 for offices (1.0 for storage) for immediate deflections and 0.25 for longterm deflections (0.5 to 0.8 for storage). The expected value of the concrete strength, not the lower-bound characteristic concrete strength, can be used in deflection calculations. There appears to be agreement that incremental deflection after construction of partitions and finishes is more critical than immediate deflection. 5,9,10 There is also general agreement that the limiting incremental deflections are span/500 for brittle partitions; otherwise, span/250.
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Table 10—Comparison of simple span beam span/ thickness ratios: incremental deflection < span/250 ACI
ρ, %
ρ′, % 318-08*
CSA A23.3†
BS Eurocode Gardner 8110-97ठ2|| and Zhang#
< 0.5
0
16
16
21.2
17
19.6
1
0
16
16
16.9
13.2
19.8
1.5
0
16
16
14.8
11.9
17.6
2
0
16
16
13.5
11.3
16.6
1.5
0.5
16
16
16.9
13.5
23.2
2.5
0.5
16
16
14.3
11.6
20.4
2.5
1
16
16
15.6
12.4
24.8
*
Steel yield stress 60,000 psi (414 MPa). Steel yield stress 400 MPa (58,000 psi). ‡ Calculated assuming steel service load stress 250 MPa (36 ksi). § Code provision written as span/effective depth-span/thickness calculated using d eff = 0.85h. || Calculated assuming steel service load stress 310 MPa (45 ksi). # For M sustained = 50% moment capacity and f ck = 30 MPa (4350 psi) (from Table 9). †
Table 9 illustrates the dependence of the limiting span/ thickness ratio on sustained moment/moment capacity ratio, concrete strength, and flexural reinforcement. Table 10 compares the deem-to-comply span-thickness ratios for simply-supported rectangular section beams for a deflection criterion of span/250. It is reassuring that all the ratios are circa span/20. It must be noted that ACI, CSA, and Gardner and Zhang 10 use span/thickness but BS 8110, AS 3600, and Eurocode 2 use span to effective depth, which are corrected to span/thickness using h = 1.18d in the table. Span to effective depth is appropriate for section strength calculations but span to thickness is more appropriate for deflection serviceability. The proposals of AS 3600-09, Gardner and Zhang,10 and Bischoff and Scanlon 12 formally accommodate incremental deflection limits other than span/250. The provisions of ACI and CSA do not accommodate the effect of compression reinforcement. The modifying factors for boundary conditions other than simply supported, given in Table 11, are similar for all proposals. Table 12 compares the limiting span/thickness ratios for the interior panels of flat slabs. Only the provisions of OfusoAsamoah and Gardner14 take account of the construction cycle–age of first/construction loading. Ofuso-Asamoah and Gardner14 assumed a form-plus-three reshores construction
ACI Structural Journal/July-August 2011
Table 11—Span/thickness factors for other than simple beams* Maximum positive moment One end continuous— discontinuous end unrestrained
2 wl n
One end continuous— other end integral with support
wl n -------14
Both ends continuous
wl n -------16
Cantilever
wl n -------2
ACI-CSA
BS 8110
AS 3600
Eurocode
Gardner and Zhang
1.2
—
1.3
1.3
1.2
1.2
—
1.3
1.3
1.3
1.4
1.3
1.5
1.5
1.4
0.5
0.35
0.4
0.4
0.35
-------11 2
2
2
*
Use midspan positive moment steel ratio from Table 9.
Table 12—Comparison flat slab interior panel span/thickness ratios without drop panels Span, m
Span, ft
ACI 318-08
CSA A23.3-04
BS 8110-97
AS 3600-09
Eurocode *
O-A&G† 7-day O-A&G† 3-day
Live load: 50 lb/ft 2 (2.4 kPa) 6.00
20
33
30
34 ‡
26.7 ‡
37‡
36.0
31.3
§
§
§
37
33.9
28.9
24.4||
37||
31.5
26.8
7.00
23
33
30
34
8.50
28
33
30
34 ||
25.6
2
Live load: 100 lb/ft (4.8 kPa) 6.00
20
33
30
33.8 ‡
25.3 ‡
37‡
33.8
28.9
7.00
23
33
30
33.5 §
24.4 §
37§
31.9
27.3
30
||
||
||
29.7
25.4
+10%
+10%
8.50
28
33
33.1
23.7
37
Edge panel thickness/interior panel thickness 10%
+10%
+20%
Drop panels—reduce slab thickness by 10% *
ρ assumed to be ρ0 /2 (minimum positive moment steel).
†
Ofosu-Asamoah and Gardner14: f c′ = 4350 psi ( f ck = 30 MPa).
‡
h assumed to be clear cover 20 mm + 10 mm bar diameter. h assumed to be clear cover 20 mm + 15 mm bar diameter.
§
||
h assumed to be clear cover 20 mm + 20 mm bar diameter.
sequence. The provisions of AS 3600-09 9 and OfusoAsamoah and Gardner14 (3-day construction cycle) are more conservative than the other proposals. All the provisions except ACI 318 and CSA A23.3 require iteration to determine the limiting deem-to-satisfy spandepth ratio either by calculating/assuming steel ratio or the member self-weights. Tables 8, 9, 10, and 12, however, can be used as design aids.
RECOMMENDATIONS The live loads for which deflections should be calculated should be clearly specified in the codes, taking note of the difference between expected live load and extreme or characteristic live load. For purposes of calculating the incremental deflections, it is suggested that the service load be calculated from the equation that follows, which is a compromise between the provisions of BS 8110-97 and AS 3600-2009. service load = D + α L where α = 0.4 for offices, apartments, etc.; and 0.8 for storage.
ACI Structural Journal/July-August 2011
For beams and one-way slabs, the deemed-to-comply minimum thicknesses given in Table 9 for incremental deflection limit of span/500 can be adopted. For incremental deflection limits other than span/500, at the same service load moment/nominal section design ultimate moment ratio, the limiting span-depth ratio is simply multiplied by 500/ required span-deflection ratio. As a first approximation, the limiting span-depth ratio is inversely proportional to the cube root of the ratio of the moment levels. Similarly it can be deduced that using a higher-yield-strength steel, which will increase the concrete stress for a given service moment/ design ultimate moment ratio, will also result in smaller permissible span-depth ratios. For other than simple spans, the modification factors suggested by Gardner and Zhang 10 (Table 11) should be used. For flanged sections where the ratio of flange breadth to web breadth exceeds 3, the values should be multiplied by 0.8. The deflection serviceability of flat slabs is determined by the loads imposed during the construction process (method and rate of construction), taking account of the concrete strength available when the construction loads are imposed, and the expected sustained service load. The appropriate serviceability criterion will depend on the location of the critical
459
location, midpanel, or column line, and can be immediate or incremental deflection, slope, or curvature. The slab thicknesses required to satisfy an exterior panel, interior column line incremental (28 to 5000 days sustained load) deflection of clear span/240 can be calculated using Eq. (11). The interior panels’ slab thicknesses should be calculated as 90% of the exterior panel thicknesses.
REFERENCES 1. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-08) and Commentary,” American Concrete Institute, Farmington Hills, MI, 2008, 473 pp. 2. CSA A23.3-04, “Design of Concrete Structures for Buildings,” Canadian Standards Association, Rexdale, ON, Canada, 2004, 358 pp. 3. ACI Committee 435, “Proposed Revisions by Committee 435 to ACI Building Code and Commentary Provisions on Deflections,” ACI J OURNAL, Proceedings V. 75, No. 6, June 1978, pp. 229-238. 4. Grossman, J.S. “Simplified Computations for Effective Moment of Inertia I e and Minimum Thickness to Avoid Deflection Computations,” ACI JOURNAL, Proceedings V. 78, No. 6, Nov.-Dec. 1981, pp. 423-439. 5. Thompson, D. P., and Scanlon, A., “Minimum Thickness Requirements for Control of Two-Way Slab Deflections,” ACI Structural Journal, V. 85, No. 1, Jan.-Feb. 1988, pp. 13-22. 6. BS 8110-1997, “Structural Use of Concrete, Part 1: Code of Practice for Design and Construction,” British Standards Institute, London, UK, 1997, 117 pp.
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7. Beeby, A. W., “Modified Proposals for Controlling Deflections by Means of Ratios of Span to Effective Depth,” Technical Report 456 (Publication 42.456), Cement and Concrete Association, UK, 1971, 19 pp. 8. EC 2-1-1 (2004), “Eurocode 2: Design of Concrete Structures– Part 1.1: General Rules and Rules for Buildings,” Management Centre, rue de Stassart, 36 B-1050, Brussels, EN 1992-1-1, 2004, 225 pp. 9. Australian Standard AS 3600-2009, “Concrete Structures,” Standards Association of Australia, North Sydney, Dec. 2009, 208 pp. 10. Gardner, N. J., and Zhang, J., “Controlling Deflection Serviceability by Span/Depth Limits and Long-Term Deflection Multipliers for Reinforced Concrete Beams,” Recent Developments in Deflection Evaluation of Concrete, SP-161, E. G. Nawy, ed., American Concrete Institute, Farmington Hills, MI, 1996, pp. 165-195. 11. Scanlon, A., and Choi, B.-S., “Evaluation of ACI 318 Minimum Thickness Requirements for One-Way Slabs,” ACI Struct ural Journal , V. 96, No. 4, July-Aug. 1999, pp. 616-621. 12. Bischoff, P. H., and Scanlon, A., “Span-Depth Ratios for One-Way Members Based on ACI 318 Deflection Limits,” ACI Struct ural Journal , V. 106, No. 5, Sept.-Oct. 2009, pp. 617-626. 13. Bischoff P. H., “Re-evaluation of Deflection Prediction for Concrete Beams Reinforced with Steel and Fiber-Reinforced Polymer Bars,” Journal of Structural Engineering, ASCE, V. 131, No. 5, pp. 752-767. 14. Ofosu-Asamoah, K., and Gardner, N. J., “Flat Slab Thickness Required to Satisfy Serviceability Including Early-Age Construction Loads,” ACI Structural Journal , V. 94, No. 6, Nov.-Dec. 1997, pp. 700-707. 15. Agarwal, R. K., and Gardner, N. J., “Form and Shore Requirements for Multi-Floor Flat Slab Buildings,” ACI J OURNAL, Proceedings V. 71, No. 11, Nov. 1974, pp. 559-569. 16. Choi, E. C. C., “Live Load in Office Buildings—Lifetime Maximum and Influence of Room Use,” Proceedings of the Institution of Civil Engineers , V. 94, Issue 3, Aug. 1992, pp. 307-314.
ACI Structural Journal/July-August 2011
