Two methods for calculating deflection are presented below, and these are based on the advice in TR58 Deflections in concrete slabs and beams8
The rigorous method for calculating deflections is the most appropriate method for determining a realistic estimate of deflection. However, it is only suitable for use with computer software. The Concrete Centre has produced a number of spreadsheets that use this method to carry out deflection calculations for a variety of slabs and beams9. These offer a cost-effective way to carry out detailed deflection calculations, and they include the ability to consider the effect of early age loading of the concrete. Figure 3 illustrates the principles of the method and shows how the factors affecting deflection are considered in the rigorous deflection calculations.
Finite element analysis may also be used to obtain estimates of deflection. In this case the principles in Figure 3 should be applied if credible results are to be obtained.
Panel 1
Determining long term elastic modulus of elasticity
Figure 3
Outline of rigorous method for calculating deflection
Calculate long-term elastic modulus, ELT from:
where
Wn = Serviceability load at stage n H = Creep coefficient at relevant loading time and duration o tM
Collate input data
■ Element dimensions and reinforcement details and arrangements from the ultimate limit state design
• Striking the formwork
• Casting the floor above
• Erection of the partitions and/or cladding
• Application of finishes
The sequence will vary from project to project
■ Concrete properties (see Table 1)
• Mean compressive strength (fcm)
• Mean tensile strength (fctm or fctm,fl)
■ Critical arrangement of actions
(or repeat the calculations for each arrangement to determine the critical case)
Assess whether the element has flexural cracking
■ Determine the critical load stage at which cracking first occurs. (See 'Cracking' on page 3)
■ Calculate the following properties:
• Creep coefficients, h (Annex B of Eurocode 2 or Figure 4)
• Long term elastic modulus, ELT (see Panel 1)
• Neutral axis depth for uncracked condition, xu (see Panel 2)
• Second moment of area for uncracked condition, Iu (see Panel 2)
• Calculate cracking moment, Mcr from:
Mct = fctmIu/(h - xu), using appropriate value for fam.
■ Does the moment at the critical load stage exceed the cracking moment?
• If yes, the element is cracked at all subsequent stages. Z = 1 - 0.5(Mcr/M)2 [5 = 0 for uncracked situation]
Use these critical values of fctm and 5 for subsequent stages.
• If no, the element will not crack at any stage.
Determine the curvature of the slab
■ When the slab is cracked calculate the following properties at the load stage being considered, using appropriate values for fctm, 5 and ELT:
• Neutral axis depth for cracked section, xc (see Panel 2)
• Second moment of area for cracked condition, Ic (see Panel 2)
■ Calculate the flexural curvature:
■ Calculate the curvature due to shrinkage strain 1 /rcs (see Panel 2)
■ Calculate the total curvature, 1/rt = 1/rfl + 1/rcs
Repeat the calculations at frequent intervals (say at 1/20 points) and integrate twice to obtain the overall deflection.
If deflection affecting cladding and/or partitions is required, repeat calculations for frequent combination and for loading at time of installation of partitions and/or cladding.
Estimate deflections:
■ Overall deflection (quasi-permanent combination)
■ Deflection affecting partitions/cladding (Frequent combination deflection less deflection at time of installation)
Table 1
Concrete properties fck MPa 20 25 28 30 32 35 40 50
Table 1
Concrete properties fck MPa 20 25 28 30 32 35 40 50
/cm = (fck + 8) |
MPa |
28 |
33 |
36 |
38 |
40 |
43 |
48 |
58 |
fctm = (0.3 fck(2/3) < C50/60 or 2.12 ln(1 + (fcm/10)) > C50/60) |
MPa |
2.21 |
2.56 |
2.77 |
2.90 |
3.02 |
3.21 |
3.51 |
4.07 |
fctm* = (0.3 fcm(2/3) < C50/60 or 1.08 ln/cm) + 0.1 > C50/60)a |
MPa |
2.77 |
3.09 |
3.27 |
3.39 |
3.51 |
3.68 |
3.96 |
4.50 |
Ecm = (22 [fcm)/10]03 |
GPa |
30.0 |
31.5 |
32.3 |
32.8 |
33.3 |
34.1 |
35.2 |
37.3 |
£c28 = (1.05 Ecm) |
GPa |
31.5 |
33.0 |
33.9 |
34.5 |
35.0 |
35.8 |
37.0 |
39.1 |
£cd,0 CEM class R, RH = 50% |
microstrain |
746 |
706 |
683 |
668 |
653 |
632 |
598 |
536 |
£cd,0 CEM class R, RH = 80% |
microstrain |
416 |
394 |
381 |
372 |
364 |
353 |
334 |
299 |
£cd,0 CEM class N, RH = 50% |
microstrain |
544 |
512 |
494 |
482 |
471 |
454 |
428 |
379 |
£cd,0 CEM class N, RH = 80% |
microstrain |
303 |
286 |
275 |
269 |
263 |
253 |
239 |
212 |
£cd,0 CEM class S, RH = 50% |
microstrain |
441 |
413 |
397 |
387 |
377 |
363 |
340 |
298 |
£cd,0 CEM class S, RH = 80% |
microstrain |
246 |
230 |
221 |
216 |
210 |
202 |
189 |
166 |
£caH |
microstrain |
25 |
38 |
45 |
50 |
55 |
63 |
75 |
100 |
a /ctm* may be used when striking at less than 7 days or where construction overload is taken into account. |
Pa ne 12
Useful Expressions for a rectangular section bh2
(a - A^V (ae - 1 )[a s(a — AAu)r + Asa (Au - dr )2]
= {[( Asffe -A Asz ( ae- l)) 2+ 2b(A\sdû-e + Axrdr( ae — 1 ))] °'5 — (Asae + Asr ( a — 1 ))}/( (a (
1 = £ Scsffe ""T +(l — ?) fcs ae "jT"
where
As = Area of tension reinforcement
As2 = Area of compression reinforcement b = Breadth of section d = Effective depth to tension reinforcement d2 = Depth to compression reinforcement h = Overall depth of section ae = Modular ratio
Figure 4
Method for determining creep coefficient ç^.to)
Figure 4
Method for determining creep coefficient ç^.to)
Simplified method
A simplified method for calculating deflection is presented in Figure 5. It is feasible to carry out these calculations by hand, and they could be used to roughly verify deflection results from computer software, or used where a computer is not available.
The major simplification is that the effects of early age loading are not considered explicitly; rather an allowance is made for their effect when calculating the cracking moment. Simplified creep factors are used and deflection from the curvature of the slab is approximated using a factor.
Figure 6
Values for K for various bending moment diagrams
Bending moment diagram m(5-
^□ocoococooooooa^
^TYYVYVYYVVinnnnn^
OCYTTTP
i-ffi I II
0.125
0.0625
0.104
rqi_2 8
0.102
15.6
Figure 5
Simplified method for calculating deflection
START
Calculate the moment, Mqp, due to quasi-permanent actions at the critical section (i.e. mid-span or at support for cantilever)
Obtain concrete properties,/cm, and Ec28 from Table 1
Calculate creep coefficient, ^(^¿o), using either Figure 4 or Annex B (in which case look-up fcm in Table 1)
1 Calculate long term elastic modulus, Eeff from: Eeff = Ec28/[1 + H (^¿o)]
2 Calculate effective modulus ratio, a from a = Es/Ef where Es is elastic modulus for reinforcement (200 GPa)
3 Calculate depth to neutral axis for uncracked condition, Xu
4 Calculate second moment of area for uncracked condition, lu
0.9 fctm/u
Calculate cracking moment, Mrr from: Mcr = -
(Note the factor 0.9 has been introduced into this method because the loading sequence is not considered)
Calculate depth to neutral axis for cracked condition, Xc and calculate second moment of area for cracked condition, lc
Calculate flexural curvature
Calculate total shrinkage strain £cs from £cs = £cd + £ca where:
ecd = kh £cd,o = Drying shrinkage strain kh = Coefficient based on notional size, see Tahle 2
ecd,o = Nominal unrestrained drying shrinkage, see Table 1
£ca = Bas(^) Eca(^) = £ca(^) for long-term deflection, see Table 1
Calculate curvature due to shrinkage strain 1/rcs (see Panel 2)
Calculate total curvature —— = Q—I- — gt,QP ge gss
Calculate quasi-permanent deflection from Kgp = KL2 ^gp where K can be obtained from Figure 6 and L is the span.
Calculate the deflection that will occur at the time of application of the load due to partitions and/or cladding.
1 Calculate the creep coefficient ((t,to), where t is the age when partition/cladding loads are applied and to is the age of striking. ((t,to) « ((^,to) B<(t,to). For Bc(t,to) refer to Figure 7, alternatively refer to Annex B of Eurocode 2.
2 Calculate the moment due to self-weight, partitions/cladding and any other loads which have been applied prior to the installation of the cladding/partition, Mpar and use in place of Mqp
3 Recalculate the section properties, curvature and hence deflection, Dpar, using ((t,to) or equivalent instead of ((^,to)
4 The approximate deflection affecting cladding and partitions is
Loading
ai ai
Table 2 Values for Kh
Figure 7
Coefficient for development of creep with time after loading
0.55
0.50
0.45
0.40
0.35
0.30
0.55
0.50
0.45
0.40
0.35
0.30
0.25
300 500
300 500
-t = 90, to = |
3 |
t = 90, to = |
7 |
- t = 60, to = |
3 |
— t = 60, to |
7 |
- t = 28, to |
3 |
t = 28, to |
7 |
Notes t = Age of concrete when partitions/cladding applied t0 = Age of concrete when struck fk = 30 (gcm = 38), however the coefficient is not particularly sensitive to concrete class
Notes t = Age of concrete when partitions/cladding applied t0 = Age of concrete when struck fk = 30 (gcm = 38), however the coefficient is not particularly sensitive to concrete class
Figure 8
Precambering of slabs
Precamber
Just before installation of partitions
Figure 8
Precambering of slabs
Precamber
Just before installation of partitions
Deflection due to, quasi-permanent combination
Deflection due to frequent combination
Deflection affecting partitions
Deflection due to, quasi-permanent combination
Deflection due to frequent combination
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