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Unformatted text preview: Chapter Four
_____________________________________________________________________ Serviceability 4.1 Serviceability Limit States of Deflection and Cracking 4.1.1 Serviceability limit states
• Review: The limit states (states at which a structure becomes unfit for its intended function) are generally divided into two groups:
1. those leading to collapse, referred to as ultimate limit states (ULS); and
2. those which disrupt the use of structures, but do not cause collapse, referred to as serviceability limit states (SLS).
• Two major serviceability limit states for RC structures:
o excessive deflection. The appearance or efficiency of a structure or any part of a structure must not be adversely
affected by deflections.
o excessive crack width. The appearance or durability of a structure or any part of a structure must not be adversely
affected by any cracking of the concrete. 155 o The serviceability limit states of excessive flexural deflection and excessive flexural cracking are the two that normally must
be considered in design.
• Other serviceability limit states that may be reached include
o Durability – this must be considered in terms of the proposed life of a structure and its conditions of exposure.
o Fire resistance – this must be considered to collapse, flame penetration and heat transfer.
o Excessive undesirable vibration – causing discomfort or alarm as well as damage.
o Fatigue – this may be considered if cyclic loading is likely.
o Special circumstances – any special requirements, such as earthquake, explosion (nuclear ~, thermal ~), impact, etc.
4.1.2 Approaches for design requirement
• Two approaches are commonly adopted for design requirements of
the serviceability limit states:
1. by deemed-to-satisfy provisions, such as detailing rules and
limiting span-depth ratios;
2. by analysis whereby the calculated values of effects of loads,
e.g. deflections and crackwidths, are compared with acceptable
values. 156 • In most cases of day-to-day practical design, the serviceability
limit state requirements are normally met by deemed-to-satisfy
provisions, such as (1) detailing rules and (2) limiting span-depth
ratios. 4.2 Service Loads
• The terms service loads and working loads refer to loads
encountered in the everyday use of structures. • Service loads are generally taken to be the specified loads without
load factors. • Load combinations
Traditionally, the load combinations for serviceability limit states
use a load factor of 1.0 on all service loads. For example, the SLS
load combination for dead, imposed and wind loads is
1.0Gk + 1.0Qk + 1.0Wk
(the USL load combination = 1.2Gk + 1.2Qk + 1.2Wk) • In most cases,
o design of reinforced concrete will be based on ultimate limit
state requirements, and
o serviceability behaviour will be considered as secondary check
to ensure satisfactory performance under working conditions. 157 4.3 Detailing Rules BS 8110 recommends simple rules to ensure that a structure has a
satisfactory serviceability and durability performance under normal
circumstances. These rules concern exposure conditions, fire resistance,
and quantity and spacing of steel reinforcement.
4.3.1 Minimum concrete mix and cover for different exposure condition
• Exposure conditions in environmental classification
Mild – protected against weather
Moderate – sheltered from severe rain or under water
Severe – exposed to severe rain or alternate wetting and drying
Very severe – exposed to sea water spray, de-icing salts or
Extreme – exposed to abrasive action • Mix and cover against corrosion
Nominal cover: The code states that the actual cover should never
be less than the nominal cover minus 5 mm, i.e.
≥ Nominal cover – 5 mm Fig. 4.3-1 Concrete cover
in a beam section 158 Table 4.3-1 Nominal cover and mix requirements for normal weight 20 mm
maximum size aggregate concrete Example: widely used in buildings Concrete mix • Cover for durability requirements (IStructE design manual)
Table 4.3-2 Cover for durability requirements for beams 159 • Cover for fire protection
Nominal cover to all reinforcement to meet a given fire resistance
period for various structural elements in a building:
Table 4.3-3 Nominal cover for fire resistance 4.3.2 Minimum dimensions of members for fire protection
Table 4.3-4 Minimum dimensions of RC members for fire resistance 160 4.3.3 Spacing of reinforcement
• Maximum spacing for cracking control
o The maximum clear spacings given in Table 4.3-5 apply to bars
in tension in beams when a maximum likely crack width of 0.3
mm is acceptable.
Table 4.3-5 Maximum clear spacing (mm) for tension bars in beams
Moment redistribution (%) fy
460 -20 -10 0 +10 +20 +30 210
210 NOTE 1. Any bar of diameter less than 0.45 times that of the largest bar in a
section must be ignored when apply these spacing.
NOTE 2. Bars adjacent to corners of a beam must not be more one-half of the
clear distance given in the table from the corner. o The maximum clear spacing between bars in slabs should not
exceed 750 mm or 3d under the specified conditions as follows:
1. If h ≤ 200 mm with high yield steel; or
2. If h ≤ 250 mm with mail yield steel; or
3. if As/bd ≤ 0.3%.
o If none of these specified conditions applies, the maximum
spacing in slabs should be taken as that given in Table 4.3-5.
o The amount of moment redistribution is unknown when using
Table 4.3-5, zero should be assumed for span moments and 15% for support moments. 161 • Minimum spacing for construction quality
o To permit concrete flow around reinforcement during construction, the minimum clear gap between bars, or group
of bars, should exceed
1. (hagg + 5 mm) horizontally, and
2. (2hagg /3) vertically
where hagg is the maximum size of the coarse aggregate.
o The gap must also exceed the bar diameter. 4.3.4 Areas of reinforcement
• Minimum area of reinforcement for cracking control
For most purposes, thermal and shrinkage cracking may be
controlled by the use of minimum reinforcement quantities.
Table 4.3-6 Minimum reinforcement areas NOTE. At least four bars with a diameter not less than 12 mm are required in
a rectangular section and six in a circular section.
162 • Maximum area of reinforcement for construction quality
The maximum areas of reinforcement are determined largely
from the practical need to achieve adequate compaction of the
concrete around reinforcement.
o For a slab or beam, longitudinal steel 1. 100 As
bh ≤ 4% each . 2. Main bars in beams are normally not less than size 16.
3. Where bars are lapped, sum of the bar sizes in a layer
must not be greater than 40% of the section breadth.
o For a column,
bh ⎧6% if cast vertically
≤ ⎨8% if cast horizontally
⎪10% at laps if either case
⎩ 4.3.5 Side face reinforcement in beams for cracking control
• When beams exceed 750 mm in depth, longitudinal bars should
be provided near side faces, as shown in Fig. 4.3-2. Fig. 4.3-2 Reinforcement at side face of beams
163 • These side face bars, which may be used in calculating the
moment of resistance, must have a diameter > √(sbbw / fy), where
sb is the bar spacing and bw the breadth of the section (or 500
mm if greater). • I.Struct.E. suggests that the size of side face bars should not be
less than 0.75√bw for high yield bars and 100√bw for mild steel
bars, where bw needs not to be assumed to be > 500 mm. 4.4 Deflection 4.4.1 Methods for deflection control
• Excessive deflections may lead to
o sagging floors, and to roofs that do not drain properly
o damaged partitions and finishes, and to other associated troubles
• BS 8110 states that the final deflection (including the effects of creep and shrinkage) should not exceed either of the following
limits (for horizontal members):
o Span/250 (in general for beams)
o Span/500 or 20 mm whichever is the lesser for brittle materials
o Span/350 or 20 mm whichever is the lesser for non-brittle partitions or finishes
• Excessive response to wind load
o Excessive accelerations that may cause discomfort or alarm to occupants should be avoided.
164 o Storey drift (relative lateral defection in any one storey), which may damage to non-structural elements, should not exceed
h/500, where h is storey height.
o For tall building design in Hong Kong, the top drift ≤ H/ 500 , where H is the total height of the structure.
• Two methods are given in BS 8110 for checking that deflection is not excessive:
1. Limiting the span/effective depth ratio – This method is used
in all normal cases (BS 8110: Part 1).
2. Calculation of deflection from the curvature of the crosssection subjected to appropriate moments (BS 8110: Part 2). In day-to-day practical design, deflections are controlled by
simply limiting the span/depth ratios. 4.4.2 Deflection control of beams by limiting the span /depth ratio (BS
8110: Part 1, clause 3.4.6)
• Basic span/effective depth ratios for beams
o The code gives a set of basic span/effective depth ratios for rectangular and flanged beams to limit the total deflection to
span/250 and the part of deflection occurring after construction
of finishes and partitions will be limited span/500 or 20 mm,
whichever is the lesser, for spans up to 10 m.
165 Table 4.4-1 o To obtain the span/effective depth ratios, the basic span/effective depth ratios should be modified according to the
actual span and amounts of reinforcement provided.
• Span/effective depth ratio = Basic span/effective depth ratio
× span modification factor
× tension reinforcement modification factor
× compression reinforcement modification factor o Span modification factor = 10/actual span (4.1)
(4.2) o Tension reinforcement modification factor = 0.55 + 477 − f s
120⎜ 0.9 + 2 ⎟
⎝ (4.3) where the service stress is estimated by
fs = 5 f y As ,req
8 As , prov ⋅ 1 βb (4.4) in which As,req is the area of tension steel required at mid-span
to support ultimate loads, and As,prov the area of tension steel
provided at mid-span (at support for a cantilever).
166 Table 4.4-2 Tension reinforcement modification factors (Clause 220.127.116.11) o Compression reinforcement modification factor 100 As , prov ⎤
⎡100 As , prov
= 1+ ⎢
)⎥ ≤ 1.5
⎦ (4.5) Table 4.4-3 Tension reinforcement modification factors (Clause 18.104.22.168) 167 4.4.3* Deflection calculations (BS 8110: Part 2, clauses 3.7)
o The difficulties concerning the calculation of deflections of concrete members arise from the uncertainties regarding:
Stiffness EI (cracked and uncracked sections)
Effects of creep and shrinkage
o In the elastic theory, the general deflection can be expressed as Δ = ∑∫ NN
ds + ∑ ∫
ds + ∑ ∫
EI o For a flexural member, such as a beam d2y
dx (Moment–area theorems express deflections in terms of the M/EI diagram.)
EI ∴ d2y 1
dx 2 r Curvature–area theorems express deflections in terms of the
The curvature–area theorems can be used for deflection
calculations even where the deflections are caused by other
effects than bending, e.g. creep and shrinkage.
o The method of calculation for deflections in BS 8110: Part 2 is based on the calculation of curvature of sections, with
allowance for creep and shrinkage effects.
* Optional course materials for CIVL 232 168 (2) Calculation of deflection from curvatures (Part 2, clause 3.7)
o Deflection: a = Kl 2 1
rb (4.6) where K is constant which depends on the shape of the bending
moment diagram in Table 4.4-4, l the effective span, and 1/ rb
the curvature at midspan of a beam or at support of a cantilever.
Table 4.4-4 Values of K for various bending moment diagrams 169 (3) Calculation of short-term curvatures
o Cracked section MR
rb E c I x (4.7) where MR – reduced moment (to be defined in Eq. 4.12)
Ec – modulus of elasticity for concrete
Ix – moment of inertia for transformed section about
For a cracked section, the stiffening effect of concrete in
tension zone is taken into account by assuming that the
concrete develops some stress in tension (tension stiffening)
represented by a triangular stress distribution (Fig. 4.4-1) Fig. 4.4-1 Tension stiffening of cracked section In BS 8110, fct = 1 N/mm2 for short–term loading
fct = 0.55 N/mm2 for long–term loading
The force of the concrete in tension
Fct = f ct (h − x )2 b
2(d − x ) (4.8) 170 The neutral axis depth can approximately be determined for the
cracked section only using the transformed area method (see
Fig. 4.4-2). In Fig. 4.4-2, the modular ratio α e = E s / Ec . Fig. 4.4-2 Cracked section − Taking moments about the neutral axis leads to
bx + α e As (x − d ′) = α e As (d − x )
2 (4.9) This is solved to give x (Fig. 4.2-3).
− The moment of inertia of the transformed section about N.A.
(axis x-x) is given (Fig. 4.2-4) by 1
I x = bx 3 + α e As (x − d ′) + α e As (d − x )
3 (4.10) − From Fig. 4.4-2, the moment of resistance of the concrete in
tension M ct = Fct ⋅ 2(h − x ) / 3 , or
M ct = f ct ⋅ (h − x )3 ⋅ b
3(d − x ) (4.11) − The reduced moment M R = M − M ct (4.12)
171 where M is the applied moment.
determined by Eq. (4.7), i.e. The curvature is then MR
rb Ec I x Fig. 4.4-3 Neutral axis depth of cracked section (d’/d = 0.1) Fig. 4.4-4 Second moment of area of cracked section (d’/d = 0.1) 172 o Uncracked section o Fig. 4.4-5 Uncracked section Referring to the transformed section in Fig, 4.4-5(a), the
equivalent area is
Ae = bh + α e ( As + As ) (4.13) The location of the centroid is found by taking moments of all
areas about the top face and dividing by Ae ,
bh 2 / 2 + α e ( As d ′ + As d )
Ae (4.14) The moment of inertia Ix about the neutral axis (x-x) is
2 bh 3
+ bh⎜ − x ⎟ + α e As (d − x ) + α e As ( x − d ′)
⎠ (4.15) The curvature is 1
rb E c I x (4.16) where M is the applied moment.
173 o The short-term curvature at a section can be calculated using assumptions set out an uncracked or cracked section. The larger
value is used for the deflection calculations, i.e. ⎧⎛ 1 ⎞
= max ⎨⎜ ⎟
⎪⎜ rb ⎟ uncracked
⎩⎝ ⎠ ⎫
⎝ b ⎠ cracked ⎪
⎭ (4) Calculation of long-term curvatures
In calculation long–term curvatures it is necessary to take into
account the effects of creep and shrinkage in addition to the
reduced tensile resistance of the cracked concrete.
o Creep (BS 8110: Part 2: Clause 7.3) Load on concrete causes an immediate elastic strain and a
long-term time-dependent strain knows as creep.
This is allowed for generally by reducing the effective
modulus of elasticity of the concrete:
E eff = Ec
1+ φ (4.17) − The creep coefficient φ is used to evaluate the effect of
− The values of φ depend on
1. age of loading
2. effective section thickness
3. ambient relative humidity
174 Fig. 4.4-6 Creep coefficients
o Shrinkage (BS 8110: Part 2, Clause 7.4) Concrete shrines as it dries and hardens. The curvature can
be caused by shrinkage. In general, more shrinkage will occur at the top because the steel area is less at top than at
bottom (Fig. 4.4-7). Fig. 4.4-7 Concrete shrinkage Curvature due to shrinkage must be estimated and added to
that due to applied moments, such that
175 1 ε csα e S s
I (4.18) where εcs – free shrinkage strain αe – modular ratio Es / Eeff
Ss – first moment of area of reinforcement about the
centroid of the cracked or gross cross–section. Fig. 4.4-8 Drying shrinkage εcs
o Total long-term curvature (BS 8110: Part 2, Clause 3.6) = long-term curvature under the permanent load
+ (short-term curvature under the total load
− short-term curvature under the permanent load)
+ shrinkage curvature (4.19) where the permanent load = Gk + 0.25Qk, which is
recommended in BS 8110: Part 2, clause 3.3. 176 (5) Procedure for deflection calculations
γm = 1, and γf = 1
moments due to: the total load (normally Gk + Qk)
the permanent load (Gk + 0.25Qk)
2. Short-term curvatures due to the total load and the permanent
for cracked section
use the greater value for uncracked section 3. Long-term curvature due to the permanent load
(considering creep effect)
4. Shrinkage curvature
5. Final curvature (total long-term curvature)
using Eq. (4.19)
6. Total long-term deflection
using Eq. (4.6): a = K l2 1
rb Examples in References
Calculation of a deflection (Text Example 6.2, pp. 119-123)
Deflection calculation for a T-beam (Reference by MacGinley and Choo,
Example 6.2, pp. 137-144) 177 ...
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This note was uploaded on 12/24/2011 for the course CIVL 232 taught by Professor Jskuang during the Spring '06 term at HKUST.
- Spring '06