How_Deflection_D11.pdf

Eurocode 2 Design Aid

Calculating deflections of concrete members

No 6 in a series of design leaflets to Eurocode 2

Why check deflections?

Deflections - overview

EC2 includes deemed-to-

satisfy span to depth ratio

m e t h o d s f o r e n s u r i n g

compliance with acceptance

criteria. These rules will be

a d e q u a t e a n d p r o v i d e

economic solutions for around

90% of designs.

In the past structures tended to be stiff with relatively short spans. As

technology and practice have advanced, more flexible structures have

resulted. There are a number of reasons for this, including:

Increases in reinforcement strength leading to less reinforcement

being required for the ultimate limit state, and resulting in higher

service stresses in the reinforcement.

However, such

methods are not intended to

predict how much a member

will deflect

Increases in concrete strength resulting from the need to improve

both durability and construction times, and leading to higher service

stresses in the concrete.

, and there can be

circumstances where the

calculation of deflections is

desirable:

A greater understanding of structural behaviour and the ability to

analyse that behaviour quickly by computer. This has led to more

slender structures built with less material.

The design of floor slabs is typically determined by the SLS. Given

that slabs constitute 80 to 90% of the cost of a concrete frame, it is

essential that they are dimensioned as economically as possible.

When specified deflection

limits are more onerous than

those recommended by the

design code, or if deflection

estimates are required by the

c l i e n t o r o t h e r d e s i g n

disciplines.

Client requirements for longer spans and greater operational

flexibility from their structures.

EC2 provide s designers with a comprehensive methodology for designing

at the SLS. The deflection calculation approach recommended by EC2 is

more open than that of BS8110. EC2 also has the advantage of being able

to account for the effects of early age construction loading, by considering

reduced concrete tensile strengths.

M o r e e c o n o m i c d e s i g n s

(smaller members) may result

f r o m a m o r e r i g o r o u s

approach.

The amount of movement to

be accommodated can have a

significant influence on the

cost of fixings for cladding and

partitions.

Required data

Many input parameters may be required before calculation can

commence, such as concrete properties ( mean compressive and tensile

strengths, w/c ratio, cement content, elastic modulus ), curing time,

ambient temperature and relative humidity.

Prediction of a member's loading history is also required in order to

calculate creep factors, other time-related concrete properties and to

determine the stage at which the member first cracks. Ages at striking

and the casting of any level above may also be needed.

There will always be a degree of uncertainty in assessing the many

necessary parameters and properties ( calculation methods are most

sensitive to values of f , E and f), unless detailed tests are carried out

ctm c

and the construction programme and propping methods are known.

Without such knowledge, calculated and measured deflections may

differ by as much as 30%.

The above should be borne in mind when advising clients, curtain

walling designers etc of expected movements.

During the early life of a

suspended floor, it is subject

to loads resulting from the

c o n s t r u c t i o n p r o c e s s .

Constructing a slab above can

cause temporary overload and

p r e m a t u r e c r a c k i n g ,

especially if a slab is designed

for low imposed loads. This

can significantly influence

later deflections, and may

need to be taken into account.

Sponsored by the

Concrete Industry

Eurocode 2 Group

(CIEG), including

BRE

Calculating deflections of concrete members

Calculating deflections - The process

ULS design

Calculate

flexural using

and equation

1/r

(7.18)

Find free shrinkage strain

from

e cs

Annex B2

based on assumed

economic slab depth,

Calculate ( both cracked

& uncracked ) and find final

value using equation

1/r cs

If M > M, section is

uncracked, otherwise

calculate

from equation

(7.18)

Calculate

Quasi-permanent M

(7.19)

at critical section

Total curvature

= +

1/r 1/r 1/r

tot

Mid-span M, or

support M for

a cantilever

Obtain

Calculate

modular ratio,

uncracked and

cracking moment,

cracked and

a e

Approximate quasi-permanent

using K factor

Concrete properties

M cr

deflection

BS 8110 Pt2 (3.7.2)

f , f and E = 1.05 E

ctm

c28

from

Table 3.1

Find and

based on 0.9f from

formulae

Creep

Repeat steps 2 to 12, but

using SLS for self weight and

cladding/partitions only

cm(t)

ctm(t)

Assess from

or calculate to

Table 3.1

Annex B

Table 3.1

deflection 12 13

Approximate affecting

cladding/partitions = -

Calculate longterm

Standard formulae for section properties

may be found on page 3 of this leaflet.

The method is applicable for hand calculation or when it is difficult to predict material

properties, environmental factors or construction programme. The

detailed

method is best suited for

computer application.

Determine basic

f and E = 1.05 E

ULS Design

for ( ) striking,

( ) casting of floor above, ( ) addition

of partitions or cladding & ( ) finishes

Loading History

concrete properties

3.1

3.1.3 (2)

based on assumed

optimum slab depth,

from Table

c28

adjusted

to clause

Is known?

or <= 7 days

(b)

(a)

Determine

when K = f /w Öb is mimimum

ctm

critical load stage

Calculate using

Annex B1

creep coefficient f

at each load stage

for each load application

Yes

(

see back page

)

f ctm

as Table 3.1

Determine

If f <= 50, f = 0.3f

ck ctm cm

If f > 50, f = 1.08.ln(f ) + 0.1

f ctm derived from

2/3

for critical stage, longterm ( quasi-permanent

& total ) and at partition/cladding addition

(

composite long-term E

f (t)

eff

ctm

At critical stage

see next page

)

(a)

If restraint is minimal,values of f closer to

ctm

(3.23) may be more appropriate.

Longterm

quasi-permanent

f ctm,fl

Calculate

modular ratio,

uncracked and

cracking moment,

cracked and

a e

I M cr

Calculate

modular ratio,

uncracked , &

cracked , & c

Affecting

partitions

Longterm

total load

a e

x I 1/r u

x I 1/r

see formulae on next page

(

)

M > M ?

M < M ?

Section is uncracked

at all stages

Calculate

using

and equation (

1/r

z cri 7.18

)

Repeat

Calculate

critical from equation ( )

Use critical value of at all stages

z crit

f ctm

7.19

Find free shrinkage strain

from

e cs

Annex B2

Calculate ( both cracked

& uncracked ) and find final

value using equation (

1/r cs

Total Load

curvature

Curvature affecting

partitions

Quasi Permanent

curvature

7.18

)

Repeat the above calculations at frequent intervals along the member and find deflections by numerical integration.

Total curvature

= +

1/r

tot

1/r 1/r

Total Load

deflection

Deflection affecting

partitions

Quasi Permanent

deflection

Standard formulae for section properties

may be found on page 3 of this leaflet.

simplified

These formulae give more realistic values if

<= 7days or construction loading considered.

Calculating deflections of concrete members

Concrete properties

(28 days)

Creep coefficients

Creep coefficient, for each component of loading

Mean concrete strength,

Calculate from equations in Annex B1 or estimate from Figure 3.1

cm f





Table 3.1

Mean axial tensile strength,

Composite creep coefficient, f eff





ctm





















ctm

comp

eff

Elastic modulus,

E = 22 (f / 10 )

0.3

Table 3.1

where E =

eff

Tangent modulus,

E = 1.05 E

c28





B.1 (3)

and w , w etc are loads (or moments) applied at different ages.

Free shrinkage strain,

e = e (drying) + e (autogenous

)

Section properties - Basic equations

(3.8)

e = from

Table 3.2 (3.9) (3.10)

Modular ratio





e = from equations

(3.11)

(3.13)

eff

A s

A s 2

If striking <= 7 days or construction overload is taken into

account, the following values of f may be more appropriate.

ctm

Reinforcement ratios and







' 

( see early age loading on back page )





5 0

6 0

Uncracked section properties

ctm



 

























(



















(



(





)(



)



Cracked section properties

ceff





































(





)

























  









































ceff

























 '













































M cr



Cracking moment



ctm

Distribution coefficient













crit







(





)

Flexural curvature

uncracked

cracked

Shrinkage curvatures and where





























csu

csc

Final shrinkage curvature







(





)

cracked

uncracked

just before installation

of partitions

Precamber

Precamber of up to 50% of the quasi-permanent

deflection is acceptable. This can effectively reduce

deflections below the horizontal, for comparison

with acceptance criteria ( normally L/250 for quasi-

permanent loading ).

precamber

total

quasi

permanent

Precamber does not, however, reduce the deflections

affecting partitions or cladding.

affecting partitions



Calculating deflections of concrete members

Early age loading

Commercial pressures are such that

there is often a requirement to strike the

formwork as soon as possible and

move onto subsequent floors with the

minimum of propping. Tests on flat

slabs have demonstrated that as much

as 70% of the loads from a newly cast

floor ( formwork, wet concrete,

construction loads ) may be carried by

the suspended floor below ( less for

other forms of construction ). Such

loads from above will vary depending

upon construction programme and

propping method, but may exceed the

design service load and have

implications for deflections ( see BCA

publication 97.505 ).

Flat slabs

Flat slabs are one of the most popular and efficient floor systems, but they are difficult

to analyse, as they require a two-way approach.

If analysed in the two orthogonal directions by spreadsheet or sub-frame analysis, there

are methods of combining results to obtain a mid-panel deflection (see section 10.3 of

main report), but these methods may not give a sufficiently reliable estimate of deflection.

Gross-section finite element and yield-line analyses, while providing good ULS solutions,

tend not to provide a reliable estimate of deflections because:

They tend to overestimate moments taken by edge and corner columns ( or

underestimate them if supports are taken as pinned ).

Gross-section analysis does not take account of reinforcement or the degree of

cracking ( unless modified section properties for each element are calculated after a

preliminary run, and re-input and run for a second or third time ).

Yield-line methods alone cannot predict SLS behaviour, and as reinforcement

patterns do not match the elastic distribution of moments, deflections and associated

crack widths may be significantly increased (see Jones).

Contemporary finite element software packages which automatically calculate all

cracked section properties and iterate the analysis to find a balanced solution are quicker

to use and offer far better deflection predictions. But care is required to ensure that the

input of materials data is appropriate.

Further improved flat slab analytical methods are likely to be available in the near future.

These should have the added advantage of providing a better model for the partial

yielding and re-distribution of moment that occur locally around supporting columns.

Columns above and below the floor should be modelled, rather than assuming pinned

supports. Also, more reliable estimates of deflection will result if the slab areas occupied

by columns are modelled as being very stiff.

Stiffnesses at ULS (reinforcement design) and at SLS (deflection/cracking) are very

different. An analysis based on an ULS or gross-section stiffness matrix cannot therefore

be used for estimating deflections.

It is therefore necessary to find the

critical loading stage ( usually at

construction overload ) at which

cracking first occurs.

The critical load stage corresponds

with the minimum value of K.

f ct



K = , where b is a duration

coefficient (always 0.5 for permanent)

and W is the stage loading.

Acceptance criteria - Flat slabs

and other two-way systems

The tensile strength, f and the

ctm

distribution factor, z associated with

this critical stage, for use in long-term

deflection calculations, can then be

fixed for use at subsequent stages ( see

flowchart on page 2 ).

If maximum permitted deflection =

then deflection at

should not be greater than

L/n

2a/n

EN 1992-1-1, Eurocode 2: Design of concrete structures - Part 1: General rules and rules for buildings ( and National Annex ) .

Webster M, Goodchild C, Webster R, Vollum R, Clarke J and Chana P, Deflections in Concrete Slabs and Beams, Concrete Society 2004.

Moss RM and Webster R, EC2 and BS8110 compared, The Structural Engineer 2004.

Narayanan RS and Beeby AW, Concise Eurocode , BCA 2004.

Narayanan RS, Goodchild C, Jones A and Webster R, Worked Examples for the design of concrete buildings, BCA 2004.

Goodchild C and Webster R, Spreadsheets for concrete design to EC2, RCC 2004.

Jones AEK, Practical assessment of the deflection of flat slabs , Concrete Communication Conference 2001, BCA 2001

In addition to these “How to Design”

leaflets, the following design aids for

EC2 are available from the Concrete Centre

Concise Eurocode

Worked Examples

Spreadsheets for Concrete Design

The suite of spreadsheets contains versions for continuous solid slabs

and beams that calculate deflections using the methods described

in this leaflet. These may also be used to obtain appropriate

values for fctm, Ec and f ( creep factor ).

How to design leaflets in this series

The Concrete Centre

Riverside House, 4 Meadows Business Park

Station Approach, Blackwater

Berkshire RG45 6YS

Concrete

$ $ $

Basis of design Beams Solid slabs Columns & walls

Flat slabs Calculation of deflections Fire design

Design for durability Foundations Detailing

Telephone: 07004 822 822

Website: www.concretecentre.com

These guides are available for free download at www.concretecentre.com/bestpractice

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: