Deflection and Crack Widths

8/4/2019 Deflection and Crack Widths

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12. Deflection and Cracking

of Beams and Slabs

Introduction

Deflection Limits

Control of deflections - beams and slabs

Comments on Cracking

University of Western Australia

School of Civil and Resource Engineering 2004


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INTRODUCTION

Today, much greater attention is now required to deflections and cracking

than in earlier years because:

higher strength materials are used - we try to use less of them!

faster construction programmes - we prestress and load structures earlier!

more critical public - community expects a higher standard!

THESE ARE SERVICEABILITY CONCERNS

EXAMINE UNDER WORKING LOADS ! ! !

Distinguish between:

Member not flexurally cracked e.g. fully prestressed member, and

cracked member - section has reduced stiffness due to cracking.


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Deflection D, at or near mid-span - may or may not be measured

from original constructed shape of member!

Crack width w, measured at extreme surface of concrete

Our objective is to limit D and w. First, deflection limits . . .

What is meant by deflection and crack width?


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DEFLECTION LIMITS

Deflection to be limited, and its magnitude, depends on the

serviceability condition being considered:

For visual effect, usually total deflection.

For cracking of partitions, usually deflection after

partitions are attached.

For bridges, usually deflection due to live load.

It is the designers responsibility to make these decisions,

and to get them right.

In all cases, the total deflection must be limited toSpan / 250

So we estimate the appropriate deflection, and ensure it is less

than the required limit. In the following, only TOTAL deflection

is examined - adjust where required . . .


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CONTROL OF DEFLECTIONS

D tot = D short term + D long termD short term : Caused by the larger of

construction load, and

short term service load G + yS Q

Dlong term :

Caused by shrinkage and creep under long

term service load G + yL Q

To estimate deflections (usually at or near the mid-

span of a beam or slab) the best guidance we have for

member stiffness is the empirical formula of Branson:

Ief= Icr + (I - Icr) (Mcr/Ms.s)3 = 0.005;

0.6I for reinforced

sections where

p < 0.005.

Well apply this formula generally for short and long

term deflection estimates, but deemed-to-comply

methods may be used to speed up our design . . .


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D short term and D long term both use Iefas estimated above.

The effects of shrinkage and creep are estimated by

another Branson formula:

kcs

= 2 - 1.2 Asc

/Ast

>= 0.8

This is used to estimate the additional long term deflection.

Now a reminder about calculating

section properties . . .

CONTROL OF DEFLECTIONS

D long = kcs . SL


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Calculation of section properties for the estimation of

deflections and crack widths:

Applies to a simple rectangularsection with one layer of rebar, subject

to working load moment, based on G + ysQ, as shown.

For more complicated sections, a similar approach is adopted.


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ULTIMATE

STRENGTH

DESIGN

1.2 g + 1.5 q

M*max

Select Ast so that fMuo >= M*max,and check ductility:

For this section, calculate

I , Icr , Mcr

DEFLECTION

CHECK

g + ySq

Ms.s

Calculate Ief D short term

g + yLq

Ms.L

Using Ief, calculate D long term usingkcs = [2 - 1.2 Asc/Ast] > = 0.8

D tot = D short term + D long term

Simply supported RC Beam

IfD tot


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Continuous RC Beam

c

Both positive and negative values of Iefmust be used in an

averaging procedure. Branson and others have shown that the

following method achieves good results:

ULTIMATE

STRENGTH

DESIGN

a b

a b

c

1.2g + 1.5q

Select Ast at critical sections

from either linear elastic, or

mmt. redistribn methods:

Asta-

Astb+ Astc

-

Iefa Iefb Iefc

Ief = 1/4 Iefa + 1/2 Iefb + 1/4 Iefc

If simple support at c:

Ief = 1/2 Iefa + 1/2 Iefb

DEFLECTION

CHECK

g + ySq, etc

Using Ief, proceed as for SS beam-some iteration may be required.

Obviously a job for a well trained computer !

A simpler method ? . . .


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2. Reinforced concrete beam - Deemed-to-comply method

Simpler, yes; but restrictive in application.

Involves ensuring that the span to depth ratio is limited to a calculated value:

Lef k1 (D/ Lef) befEcd k2 F d.ef

[ ]= 0.005

0.1 - 13.5p for p < 0.005

Lef= {L ; Ln+ D] min

Fd.ef= (1+ kcs )g + (yS + kcs yL )q

and Slabs? . . .


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3. Reinforced concrete slabs, edge-supported panels -

Simplified calculation

A one-way slab, simple or continuous, is best treated just like a beam.

A two-way slab, with edge supported panels, is treated thus:

Lx Lx Lx

Equivalent

beam of 1 metre

width, spanning

in short

direction

Apply a UDL to the equivalent beam, the load being a proportion of the UDL to

which the slab is subjected. The proportion is given by:

Ly4/ (a Lx4 + Ly4) and proceed as for a beam with the same support constraints.

a from Table 9.3.4.2 of AS3600 Simpler method ? . . .


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4. Reinforced concrete slabs, edge-supported panels -

Deemed to comply method

Simpler, but more restrictive in its application.

Similar to deemed-to-comply method for beams.

Lef (D/ Lef) Ecd F d.ef

[]


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5. Reinforced concrete flat slabs: Simplified calculation

The Code refers to the idealized frame method of analysis - not treated inthis course. For slabs where the span in the two directions direction do not

differ by more than 10%, the following course is acceptable.

Reasonable accuracy can be achieved for deflection calculations by treating

the slab as orthogonal one-way slab, calculating the deflection along the

centre-line of the column strip in one direction, then along the middle stripin the other direction. The method is outlined in Warner et al.

For preliminary calculation, the deemed-to-comply method may also be

used, but should be checked when sufficient design data is available.

Note that this is still the subject of research. Seek advice before applying.

Now for prestressed members . . .


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Prestressed Beams and Slabs

We have already dealt with afully prestressedmember.

There are two possible cases for apartially prestressedmember:

CASE 1: Under sustained load G + yLQ cracks are tightly held

closedby the prestress force; and

CASE 2 : Under sustained load G + yLQ cracks are not held closed.

In this course, we are concerned with Case 1 only.

Bransons method may be used for calculating Ief, but with a couple of

modifications, demonstrated by Branson himself :

Mcr is replaced by Mcr = Mcr - Mbal = Mcr - Pe, and

Ms is replaced by Ms = Ms - Mbal = Mcr - Pe

With these modifications, the methods for reinforced slabs may be adopted.

Typically, the calculations are somewhat simpler.

This modified method is best shown graphically . . .


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Note that D is the deflection measured from the

balanced condition. The actual deflection calculation

must be modified to allow for the pre-existing

deflection.

Bransons Modifications for Prestressed Members

Cracking . . .


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No specific guidance on limits is provided in code.

ACI recommendations are commonly followed today:

Exposure Condition Maximum allowable crack width

Dry air or protective membrane 0.4 mm

Humid , moist air, soils 0.3 mm

De-icing chemicals 0.2 mm

Seawater and sea water spray 0.15 mm

Wetting and drying 0.15 mm

Water retaining structures 0.1 mm

So for most structures, 0.3 mm, or possibly 0.4 mm, is the limit we should try

to achieve. For special cases, greater attention is required.

AS3600 is directed to 0.3 mm for external applications.

COMMENTS ON CRACKING


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Cracks occur when the tensile strength of

concete is exceeded.

This may occur due to:

flexural (bending) action; or

restraint to shrinkage and creep; or

combination of the above.

Crack width depends on

spacing of cracks; and

mean strain in concrete between cracks; and

rebar edge distance and spacing.

Is section cracked? i.e. is Ms > Mcr?

Is restraint significant, and if so is

rebar area adequate to control

crack width?

Depends on size of rebar db

Depends on stress in rebar fs

We have dealt with this topic in SCD322.

Just some reminders . . .


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8.6.1(b)

8.6.1(ii)

8.6.1

8.6.1(iv)

Table 8.6.1(A) Table 8.6.1(B)

FINISH

Is fscr.1 < 400 MPa?

yesyes

yes

yes

yes

no

no

no No actionrequired

Is MG + MQ > Mcr?

where Mcr is calculated

for 3.0 MPa tensile

strength?

Ensure nearest bar distance

< 100 mm, and bar spacing

< 300 mm

Is beam fully

enclosed?

No action

required

Adjust to

comply

Is Ast>1.8Act/fs, where fs =

{-173loge(db)+760MPa;

500 MPa}min ?

8.6.1(i)

Uncracked

section

Bar

spacing

Is fscr < -

173loge(db)+760MPa?

no

Nearest

bar

distance

Act

Amend

design

Is fscr < -0.8(spacing)+400

MPa?

no

no

BEAM IN

FLEXURE

Table 8.6.1(A)


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Slabs - Control of shrinkage induced cracking

For slabs, we are concerned to limit the width of cracks due to restrained

shrinkage. For this condition, various options are available to the designer,

who must determine the degree of crack control appropriate to the design case.

The categories are:

minor control- intended where slab is interior, and where cracks will not

provided a problem and are not visible.

moderate control- intended where cracking is visibly acceptable, and does

not cause waterproofing or durability concerns.

strong control- intended where cracking is visibly offensive, or where

waterproofing or durability concerns are present.

Note how the area of steel required is

diminished by any prestress which exists.


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SUMMARY

Checking of deflections and crack widths is essential in modern design.

Guidance is provided on acceptable deflections, but the designer must

ensure that the structure is suitable for its intended service.

Deflection calculation procedures are provided for beams and slabs.

Some concern exists about the estimation of the effects of shrinkage-induced stresses. Use Mcr = Z. 0.6 (fc)

0.5 until resolved.

A deemed-to-comply crack width procedure is provided. Use with

care to ensure that all the requirements are covered. The procedure is

directed to 0.3 mm width for external members (may not be adequate),

and about 0.45 mm for internal members (may also not be adequate).

Care to ensure minimum steel is provided in slabs, appropriate to the

application.

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Deflection and Crack Widths