waffle slabs. estimation of deflections in waffle slabs · waffle slabs 3 estimation of deflections...

ESTIMATION OF DEFLECTIONS IN WAFFLE SLABS

[From the publication Los forjados reticulares: diseño, análisis, construcción y patología, by Dr. Ing. Florentino Regalado Tesoro. Biblioteca Técnica CYPE Ingenieros, S.A. (Avda. Eusebio Sempere, 5; 03003 ALICANTE; Spain). ISBN 84-930696-5-5. 1st edition, 2003. Section 10.9, pages. 402-409]

The deflections that are usually relatively simple to evaluate, as far as this is

possible, in reinforced concrete structures are the instant deflections at low load levels, since these are situations in which the cracking levels are relatively moderate and the general behaviour of the structure has a generally sufficient level of elasticity.

As phenomena such as shrinkage and creep become involved, and cracking

levels increase, the precision in the evaluation of delayed deflections becomes quite a problem, and one needs to be satisfied with a sufficient estimation of that precision that will allow the pieces that are designed to be positioned and construction to be performed at tolerable levels of deformation, even accepting errors of a certain magnitude in this evaluation, which some authors estimate at between 25 and 50%.

Our personal experience in the estimation of slab deflections in building

construction, which was subsequently verified in load trials, indicates that the simplified estimation criteria that are customarily used have a conservative character. The deflections measured during the load trials were found to be considerably smaller than the ones estimated in the calculations, since many factors that cannot be parameterised are not taken into account. In our tests on waffle slabs, it was found that the construction of a waffle slab with non-reusable concrete blocks, compared to that of a waffle slab built with lightening polystyrene blocks, leads to instant deflections that are about 1.25 times smaller, though both slabs exhibited identical structural sections.

• The manual method that allows an estimation to be obtained of the deflections

in waffle slabs was developed by Scanlon and Murray, and has been endorsed by the American Concrete Institute (ACI) since their authors published it in 1982.

The method starts by using the schemes that are established in the virtual

gantries method, after Figure 10.10, and, considering the bands in this method, it obtains fictitious loads q1 and q2, which correspond hypothetically to the different bands as a function of the moments assigned to these, based on the moments provided by the general calculation of the gantries or, when these are unavailable, on the estimation of these moments by the DIRECT CALCULATION METHOD.

Waffle slabs 1 Estimation of Deflections in Waffle slabs

Fig. 10.10. Band scheme by Scanlon and Murray for the calculation of deflections.

To calculate deflections, it is recommended to consider the width corresponding

exclusively to the abacuses as supports band. The comprehensive moments of the virtual gantry, which may be considered as a reference, are shown in the following scheme.

Fig. 10.11. Direct reference moments that can be manuallytaken into account in the deflection calculations.

Instant deflection

If one wishes to obtain the instant deflection at point 5, for a uniformly

distributed load P1/m2, one obtains, first, the fictitious load fraction (qly), which is assigned to the supports band in the y-direction, for example; and the fraction (q2y) which corresponds to the centre band also in this direction; then doing the same in the x-direction.

( )

( )

8Lq

M0.4 2

MDMI0.258

Lq M0.6

2MDMI0.75

2y2y

vyyy

2y1y

vyyy

⋅=⋅+

+⋅

⋅=⋅+

+⋅

where: q1y: y-supports band load Ly: clear span according to y q2y: y-centre band load Ly: clear span according to y



nce the loads have been obtained, the deflections of each band at the mid-point can be calculated, as if ordinary beams were involved, first in the y-direction and then in the x-direction, operating with the Branson equivalent inertia (according to EHE).

O

( )

( )eycC

2yyy

eycC

4y2y

5y

eysC

2yyy

eysC

4y1y

1y

IE61LMDMI0.25

- IE384

Lq5 f

IE61LMDMI0.75

- IE384

Lq5 f

⋅⋅

⋅+⋅

⋅⋅

⋅⋅=

⋅⋅

⋅+⋅

⋅⋅

⋅⋅=

If the gantry geometries and layout are regular, the deflection obtained in 1

according to y may be considered identical to that which could also be obtained in 2 according to y; and if they were square panels, they would also be quite similar to the deflection in point 4 and point 3.

Should that not be so, one would have no choice but to establish the virtual

gantries and to operate in the way indicated, obtaining those deflections in both directions.

The instant deflection at point 5 would be given by:

2f f (end) f

f2

ff f

f2

ff f 2

515

5

5y4x3x2

5

5x2y1y1

5 +=

⎪⎭

⎪⎬

⎫

++

=

++

=

The inertia moments that need to be introduced into the formulas to calculate the

deflections are the inertia equivalent moments given by Branson. The sections that need to be considered in each analysed band differ from each

other when it comes to obtaining the equivalent inertia moments. At the ends of the supports bands it will be necessary to consider the rectangular

abacus sections; and in the centre spans of these bands, a set of I-ribs with reinforcing in the bottom part of the cores.

The I-ribs are considered in the calculations of the centre band deflections, but

one must bear in mind that, at the ends, the reinforcement lies in the top part, whereas in the spans, the reinforcement lies the bottom part.

Fig. 10.12. Position of the reinforcemencalculating the cracked inertia

t when it comes to s in the bands outside the

abacuses.


f

3

a

fb

a

fe

IMM-1I

MM I

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛=

1

bflexct,f y

If M

⋅=

where: Ie: equivalent inertia moment in each section of the given bands. If Mf > Ma one shall set Ie = Ib.

Mf: cracking moment of the section.

section, in which the reinforcements may be made to

fct,flex: concrete strength under bending tension. y1: distance from the section centre of gravity to the farthest tensile fibre. Ma: maximum flector moment applied in the life of the piece, up to the time

when the deflection is calculated, in the corresponding sections, Ib: gross inertia of the

play a role, based on the equivalency factor of steel in concrete:

C

S

EE

n = (n = 8 in normal cases)

Ec: concrete modulus of deformation for instant loads at the age that the

If: Inertia moment of the cracked section homogenised to values of concrete, in relation to the axis that passes through its centre of gravity.

After calculation of the equivalent inertia moments at the ends and in the mid-

point of the span, where the maximum moment of positive bending lies, with a view to at one should

perate

Delayed deflection

the stretch being analysed due to this ad state may be estimated by the comprehensive coefficients method established by

the EHE. This method consists of calculating the factor λ in the centre section of the span

being analysed, or in the beginning section if an overhang is involved.

deflections are evaluated (see Table 10.4)

operating with a single inertia for each stretch, Branson proposes tho with a generalised average inertia that, for continuous stretches of building construction structures, is given by:

Ie~average = 0.50·Ie~span + 0.25·(Ie~left end + Ie~right end)

and for overhangs, conservatively:

Ie~average = Ie~overhang starting section

Once the instant deflection for a fixed load state P1/m2 has been obtained, the

additional delayed deflection that can take place inlo

ρ501 ′⋅+metric quantity of the compression reinforcementection.

ξ λ =

where: ρ': geo in the reference useful s


J. Calavera fixes the value ρ and proposes that it should be calculated with an average value given by:

ρ'm = 0.70·ρ' + 0.15·(ρ'left end + ρ'right end) in order to be consistent with the Branson formulations.

ξ is the coefficient that depends on load duration and tak

5 o1 y6 m3 m1 m2 w Onc alculated by

multiplying the instant deflection by this v

Total defle The en time in the

analysed piece is the sum of the instant deflection and the delayed deflection.

deflection, i.e. the deflection that occurs after a given time and that can affect the elements constructed (usually partition walls) after this time, can only

mplified and estimative form, though it can obviously fail, as indeed happens on

In addition, it further needs structed first, whether e flooring or the partitions. Even though the usual procedure is first to build the

artition walls, this is not always done, especially when it is sought to anticipate that ch partitions, as quite frequently occurs.

Load caused by the partition walls. Load caused by the flooring.

es into account the shrinkage and creep that occur in the pieces. It adopts the following values:

r more years: 2 ear: 1.4 onths: 1.2 ononth: 0.7 eeks: 0.5

ths: 1

e the value of λ has been obtained, the delayed deflection is calue.

ction

estimation of the total deflection that may be expected at a giv

Active deflection

The evaluation of the active

be performed if a construction program is established, albeit in a sinumerous occasions.

to be established what is con

thpusers may relocate su

As a result, generally, four load fractions need to be established: • Load caused by the weight of the structure. • • • Load caused by the service overload.


It deserves to be noted that, in the calculations of the inertia moments, one needs to act at every moment with the maximum flector moment applied in each section up to that time, since those mo reductions that occur as

as may be observed below in Chapter 12 on the construction of the structures.

ing m ans the maximum moments will be those that are almost certainly due to the construc n process, and as an order of magnitude, 1.9 times the own weight of the slabs. The values that professor Calavera provides of 2.25 for two storeys with installed formwork, 2.36 for three such storeys and 2.43 for four, as multipl actors of the own slab weight are, in our view and as explained in Chapter 12, exc

As a result, for all practical purposes and with a view to not complicating atters unnecessarily, in building structures it will suffice to calculate the Branson

g the Ma moments that are provided by the usual calculation, since in general the following is obeyed:

d

If both values are not appreciably similar, one needs to operate with the larger lease note!), with the moments resulting in a service situation, i.e.

without increasing them by the factor γf.

hus, the calculation of delayed deflection, assuming that the partitions are constru

on walls: fit calculation of the instant deflection due to the flooring: fis

d: fisu calculation of the delayed deflection of the slab own weight from

alls: fdpp

ments are intimately tied to the inertia a result of cracking in the sections.

The maximum bending moments are very likely to occur during the construction

process owing to the installation of formwork for successive storeys,

The forego e that

tio

ying fessive.

minertias, introducin

1.9 own weight ≈ own weight + permanent overload + service overloa

value, however (p

If the problem can be simplified in the manner indicated, the deflections that are

successively produced can be summed, since one is operating with a constant conservative Ie, all the calculations resulting absolutely linear.

Tcted before the flooring is installed, may be performed as follows: • calculation of the instant deflection of the own weight: fipp • calculation of the instant deflection due to the partiti• • calculation of the instant deflection due to the service overloa•

the construction deflection of the partition w

ippdpp

m

ct

fλf501

=+

ξ(

ρ)ξ(t-)

⋅′⋅

∞=λ

calculation of the delayed deflection due to the partition loads: fdt •

it1dt

m

ct1

fλfρ501

)ξ(t-)ξ( λ

⋅=′⋅+

∞=

• delayed deflection due to the load after construction of the flooring: fds


ds2ds

m

cs2

fλfρ501

)ξ(t-)ξ( λ

⋅=′⋅+

∞=

delayed deflection of the service overload fraction (persons and

dsu

• furniture) from the time that the building enters into service, which is estimated to act in a permanent way (estimation of 500 N/m2 in building construction could be a value to bear in mind): f

isu3dsu

m3

fλfρ501

λ

⋅=′⋅+

Active deflection: fit + fis + fisu + fdpp + fdt + fds + fdsu

It is reas

su=

onable and prudent, in a certain way, to count on the instant deflections that partitions produce, partic

, cl m), the

ethod that professor José L. de Miguel proposes, taken from his work on the analysis of defle

instant deflection, i.e. that: fi = fipp + fit + fis + fisu, an esthe fact

)ξ(t-)ξ(∞

ularly with the construction processes followed in Spain, since partitions are constructed in an ascending mode. In those cases in which partitions are constructed in a descending mode, the instant deflection that they produce in the delayed deflection calculation can be disregarded.

For rapid estimations and in order to perceive whether the active deflection is farose to, or goes beyond the limits recommended in the EHE (L/400 or 1 cfrom

mctions, may be of great usefulness. Based on the calculation of total

timation may be made of the active deflection (factive) by just multiplying fi by or φact taken from Table 10.5.

factive = φact · fi

Deflection in relation to the instant deflection of total load (kc = 1.05)

Flooring before partitions Partitions before flooring Building work pace town/tfloor/twall/tservice (months)

Slow 1/4/9/18

Normal 1/3/6/12

Rapid 1/2/3/6

Slow 1/4/9/18

Normal Rapid 1/3/6/12 1/2/3/6

Active deflection q tct own+floor+wall+service

(kp/m2) Coefficient φa *ct

1.15 580 200+080+100+200 0.76 0.84 0.94 1.01 1.05 1.16 650

700 720 800 900 950

1000 1100

250+100+100+200 280+120+100+200 300+120+100+200 300+150+050+300 300+150+050+400 300+150+100+300 300+150+100+450 400+150+100+450

0.75 0.74 0.73 0.77 0.72 0.77 0.80 0.78

0.83 0.82 0.82 0.84 0.86 0.84 0.86 0.85

0.93 0.93 0.93 0.94 0.94 0.94 0.95 0.95

1.01 1.02 1.02 1.05 1.04 1.02 1.03 1.02

1.05 1.07 1.17

1.18 1.06 1.19 1.09

1.08 1.17 1.16 1.06

1.07 1.05

1.15 1.14

Detectable cumulative deflection Coefficient φacm (except instant deflection of own weight) 1.20 1.40 1. 0 5 1.20 1.40 1.50 * Column recommended by F. Regalado.

Table 10.5. Coefficients for estimating active deflection starting with instant deflection (J. L. de Miguel)



Thus, for example, if the calculated instant deflection is 5 mm and the building work pace is normal, the partition walls being constructed before the flooring, for a total load of 8 kN/m2, one might expect an active deflection of the order of:

g the

However, if the obtained active deflection were very close to 10 mm, whether

above or below, it would be desirable to perform the calculations established for its valuation, in order to able to verify it.

by J. Luis de Miguel provides for φact are

box has been fi e lection is going to be calculated by mea of adesired point (C.R.) is chosen, which usually lies at the mid-point of a diagonal alig nt bet r exa of A e pr is then aske xhi l the displacem t has rgone. Summ ll th tical acem giv e gros at the t co red ) has rgon

δC pp + δsu pp: weig overload; su: service overload

oing epres ive point of the column lds δ δB.

ffic pre flection, for point

factive = 1.09·5 = 5.45 mm

Since this deflection is very far from the tolerated 10 mm, no matter how lar e

error, it would not be necessary to perform additional calculations.

e

lthough the values that the method proposedA somewhat low, the method is particularly interesting for the estimation of the

deflections that can be obtained by the CYPECAD calculation software (version 2000.1). Once the structure has been calculated, this program allows the vertical displacements at any point of the slab to be determined, for each of the hypotheses that are analysed.

Fig. 10.13. Schematic drawing to evaluate the deflection

Once the xed wher the def

ns zoom, the program is asked to display the calculation mesh used and the

nme ween columns, fo mple B; th ogram d to e bit alents that this poin unde ing a e ver displ ents

es th s displacement th poin nside (C.R. unde e. R = δ δsp +

own ht; sp: permanent

D the same with a r entat s yie A and The estimation of a ‘su

C.R., is given by: iently cise’ instant elastic de

s by the model used in the CYPECAD software (version 2000.1).

1.60β1.15 δβ f2δδ - δ δ

iiiCR

BACR

≤≤⋅=

+=

βi being an amplifying factor th lab analysed and the degree of cracking that takes place in the

CYPECAD, designed fundamentally to obtain

ng into account the singularity noted in Fig. 10.13, allow them to be converted very rapidly to deflections with notable precision, as we were able to demonstrate in Chapter 6, verifying the outcomes.

It should not be forgotten that the displacements of the nodes in a spatial

analysis are intimately related to each other, and the deflections represent differential displacements relative to the columns, while these in turn, depending on the number of storeys involved, also undergo shortening to a greater or lesser degree, though this may be disregarded for the ordinary cases of housing.

The elastic deflections that are thus obtained from CYPECAD in the mass

concrete slabs are correct and it would suffice to multiply them by a small factor of the order of βi = 1.15 to βi = 1.25 to allow them to be considered as sufficiently precise instant defle during their

tion y this factor, it would suffice to eep in mind what has been set out previously on delayed deflection, following the

compre

s, lead to smaller elastic vertical deformati strains and that, therefore, they

ne considered previously for mass concrete slab

only the questions relative to the stiffnesses of the elements of the discretised slab us

ying factor of βi = 1.30 for the δ values that the software yields, and the elastic outcomes would be sufficiently good and reliable; however, since there are also other issues of a certain importance, such as the stiffness losses that occur during

at depends on the type of sconstruction process.

Applying Table 10.5 allows an approximate evaluation of the active deflection

to be obtained. fact in CR = φact·fiCR

However, it may be noted once again that the discretisation and stiffnesses

considered in the calculation model used by the most constructive possible precise stresses and reinforcements in the slabs,

provide (with their linear elastic analysis) vertical displacements in the slabs that require a certain interpretation and processing in order to be reliable tools: we are referring to the δCR values.

If the decks are mass concrete slabs, the displacements that the software

provide , takis

ctions, bearing in mind the stiffness losses that can occur process. After they have been multiplied bconstruc

khensive coefficients method of the EHE or the simplified process of Table 10.5

by J. L. de Miguel in order to estimate the design active deflection in mass concrete slabs.

The problem becomes somewhat more complicated when one works with waffle

slabs, in which the constant stiffnesses that CYPECAD uses, for such slabons than the actual

should be treated with a somewhat larger βi factor than the os.

Ifed in the CYPECAD model were to be taken into account, it would suffice to

consider an amplif


the construction process as a result of cracking, etc., this factor needs to be somewhat larger.

es.

and shrinkage, as e uction process has on deflecti ns, cracking the sections t occur during that process.

e buildings with three storeys were calculated by CYPECAD, though without any type of simplif

• With a view to estimating a factor β (in the waffle slabs) in the most precise possible way, we have used the PhD thesis of Luis García Dutari and have compared the outcomes of his model with ours, in a series of cas

The L. G. Dutari model is based on finite elements and takes into account

cracking, creep, w ll as the effect that the constrwith the overloads thao

This model has been used to calculate three types of buildings with three spans

by three spans, and bearing distances according to the X-axis of 6, 7.50, and 9 metres, and according to the Y-axis of 6 metres. The columns are 3 metres tall and perfect embedding is assumed at the opposite end to the slab.

The sam

ication owing to the symmetries, and considering the intermediate slab as a comparative reference.

The respective edges are 20+5, 25+5, and 30+5.

Data of the analysed buildings (H-25 and B-400-S)

Spans Edge P.P. kN/m2

6x6 20+5 3.72 6x7.5 25+5 4.44 6x9 30+5 5.23

S.P.

kN/m2 S.U. Columns H = 3 m

2 2 45x45 2 2 45x55 2 2 45x65

Table 10.6

Fig. 10.14. Partial outcomes provided directly by CYPECAD forthe own weight hypothesis in the 6 x 7.50 m model.


Table 10.7 presents the results of the analyses and calculations made with the CYPECAD software and the L. G. Dutari models. In general, it may be stated that the CYPECAD software provides total consistency in its results, whereas the finite element models and the simplified discretisation used by Dutari fail to establish the symmetries in the structure and display certain inconsistencies in some outcomes.

Values produced by the finite element model of J.L. García Dutari.

Construction pace / rapid and slow

Instant deflection (mm)

Total deflection (mm)

Active deflection (mm)

Values of δ provided by CYPECAD (mm) (See Fig. 10.14)

Type R.E. R.M.X. R.M.Y. R.C. R.E. R.M.X. R.M.Y. R.C. R.E. R.M.X. R.M.Y. R.C. R.E. R.M.X. R.M.Y. R.C.

6.66 6.66 5.83 5.88 17.08 17.00 15.42 14.71 8.33 10.00 9.17 8.536x6 4.50 4.00 3.90 3.35

7.08 6.47 6.33 6.47 16.67 16.76 15.67 14.12 8.75 8.53 8.33 7.35

7.90 8.33 7.92 7.06 19.17 20.67 18.33 17.06 10.00 12.00 10 8.247.5x6 4.75 4.00 4.40 3.60

7.08 7.06 7.00 5.88 16.67 17.65 17.00 14.71 9.17 9.12 8.67 7.06

12.50 11.00 10.42 8.82 25.00 23.33 21.25 17.65 14.17 13.00 11.25 8.829x6 5.40 4.50 5.30 4.30

10.42 10 10 8.24 21.67 20 18.67 16.18 10.42 10 8.67 7.65

Table 10.7

owever, the comp ting with average values,

allows certain practical val tor, which, on multiplying them by the results of the vertical software, version 2000.1, provides for waff , enab nstant, d active deflections to be estimated with reasonably good precision.

However, it should be noted that the values of very likely to be somewhat

conservative and, therefore, the deduced deflections are larger than the actual ones, ecause the method used by Dutari cracks the slabs more than they actually do, when

they are excessively overlrelating to successive in f storey formwork, while the values that can be deduced from the applicatio f the J.L. ied criterion more closely match reality.

Operating summary: δ: Vertical displacement of a point in the waffle slab mesh calculated by the

CYPECAD software.

Deflection = β·δ - Instant deflection βi = 1.60 - Maximum long term deflection βm = 4.00 - Active deflection βa = 2.20 An example may clarify the operating process to be followed: Let there be a 6x6

waffle slab with 20+5 edge, with dead loads of 2 kN/m2 and service overload also of 2 kN/m2, in which the deflection in a corner span is to be calculated.

H arison of the outcomes, operaues to β fac be deduced for the

deformations δ that the CYPECAD otal, le slabs le the i t an

β are

boaded during the construction process, as a result of issues

stallations on o de Miguel’s simplif



•

Since the column shorten leaves us with an initial deformation of δ =

• Owing to the discretisation performed and le cracking of the slab, the real instant deflection that may occur in the corner box is:

.

o in a . e , a e n t o expected in the bo n e

Active deflection (J. L. de Miguel) = F a

n w r cActiv le on ut δ 2. m

• And the maximum long term deflection

er to determine the deflections in the waffle slabs, the above will provide ‘relatively precise’ values; however, for simple building constru

eflection is evaluated by the virtual gantries method, given the extremely conservative character of the outcomes, these may be made more precise by multiplying

deflections by finite elements, taking into account shrinkage and creep, leads to eflection values between 30 and 60% below those obtained by the virtual gantries

method

Using the CYPECAD software to calculate the building, let us take the second storey slab and check out the vertical displacements of the two hypotheses.

Own weight 3.5026 mm Overload 1.0258 mm Total 4.52284 mm

• ing in a three-storey building is negligible, this 4.5 mm.

to t ossibhe p

Total instant deflection = βi · δ = 1.6 · 4.5 = 7 2 mm

• F llow g T ble 10.5 of J L. d Miguel the ctive defl ctio tha is t bex me tion d is:

i · φ ct = 7.2 · 1.18 = 8.5 mm

A d if e di ectly apply the orresponding b factor: e def cti (D ari) = βi · = 20 · 4.5 = 9.9 m

Maximum deflection = βm · δ = 4 · 4.5 = 18 mm

Final comment We recognise and doubt that, in ord

ction project designs, and against what is currently occurring, namely that no estimation at all is being made of waffle slab deformations, everything being left to appropriate selection of the edges and the experience of the designer, we can congratulate ourselves on having two instruments that allow us approximately to resolve the problem, albeit just that: approximately.

If the d

them by 0.5 or 0.6.

According to García Dutari and J. Calavera, a more precise calculation of the

d.

waffle slabs. estimation of deflections in waffle slabs · waffle slabs 3 estimation of deflections...

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