Reinforced Concrete Design: A Practical Approach, Chapter 12 [3 ed.] 1323496556, 9781323496558

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Citation preview

Design of Two-Way Slabs

• LEARNING OUTCOMES After reading this chapter, you shauld be able to • describe the key features of reinforced concrete two-way slabs • design two-way slabs for flexure in accordance with the CSA A2l.l desigr procedures • evaluate shear resistance of two-way slabs considering one-way and two-way shea: effects • estimate immediate and long-term deflections in two-way slabs according to CSA A23.3 requirements

~ INTRODUCTION This chapter builds on the [undamcntals presented in Chapters IO and 11, which explain,J how mulliplc span slabs and beams can be designed as continuous structures by the sir.lie· gic placcmcnl or reinforcing steel in the top and bollom regions. Continuous rcinfon:o! concrete structures that span over several suppons arc characterized by greater ncxural s1iffncss than single span structures with the same overall length. This could resull in rc p·.an ri C Vt('.'1 '. I. : ., :50 n1et_ l srctrnr1,

·~Ii:

?.,'

vert1cc1

C'

I

dr~d

nar,HJ. '1:w; u11dr.r biirl(' '.,

!,/,.--

Slab panel

P!an a)

b)

c)

CHAPTER 12

582

Figure 12.1

(cont.)

e)

12.2.3

Flat Slabs The span-to-thickness ralio !'or a llat plate ls u~ually limited b) permitted long-tmn ~~· flcctions. Thin slahs could also haYc punching shear issues in column regions. The pur.~~ing shear capacity can be increased hy thickening the slab locally around the columns.', way or drop panels. A llat plate S) stem with drop panels is called ajlar slab, as sho"n ::. Figure l 2.2a. Drop panels cause an increase in the slab stiITncss in regions subjected tor.~gativc bending moments, and therefore help to control dcllcctions in midspan regions,\ smaller amount of top reinforcement is required in slabs with drop panels than in llat pl~tr~ Drop panels arc usually square in plan, with plan dimensions typically one-thinlolt< span length in each direction or the slab. Thickness or a drop panel is often governed'''" form work dimensions (e.g. 89 mm thickness can be speciricd when 2" x 4" plank is IL

DESIGN Of TWO-WAY SLABS

583 flat slab system:_ ,.. '.e 12.2 . view and a vertical

~~·~. isometnl~imn cc1pital; c) a flat ,-i:cn: b) c~tals, and d) a flat ·" ,1th cap J. ''. olumn ca pi'tals ' and drop ~t ,1 ,th ( 1

reiS

,

'.::.J"aB'leV/-

Drop panel __ /

b)

584

Figure 12.3 Waffle slab system: a) plan and a vertical section, and b) en example of a building application.

CHAPTER 12

uuuuuuuuuuuuu

JDDDDDDDDDDDDO[ JD • DDDDDDDD • [ JD DDDDDDDD [ JDDDDDDDDDDDDO[ JDDDDDDDDDDDDO[ JDDDDDDDDDDDDO[ JDDDDDDDDDDDDD[ JDDDDDDDDDDDDD[ JDDDDDDDDDDDDD[ JDDDDDDDDDDDDD[ JDDDDDDDDDDDDD[ JD • DDDDDDDD • [ JD DDDDDDDD [ JDDDDDDDDDDDDD[ nnnnnnnnnnnnn Plan

Section a)

b)

DESIGN OF TWO -WAY SLABS

585

ure J2,4 Slab with beams: ~pical plans and vertical tions. and b) an example of :ding application /a~a srzev).

11 I I

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11

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1 : 1

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~:

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Plan

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Column

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Section a)

Section

Warllc slabs could offer economical design solu1ion for sp,ins in the range of 10 to 12 m, depending on the availab1hly ol lomung pans and rclc\'anl constnuction skills. Slab thickness varies from 75 10 130 mm boscd on either lire resistance requirements or structural considerations. Note that a 130 mm slab is needed to meet 2-hour lire rating requirements. Joist width is 150 or 200 mm, and the depth mngcs from 200 lo 6(Xl mm underside of the slab. The spacing of joists is go,·erncd by 1hc dome dimensions and code requirements. Maximum joist spacing is 750 mm. Solid slab portions (heads) arc provided for increased shear strength al the column locations. For design purposes. wafllc slabs arc considered ns flat slabs with the solid heads acting as drop panels. Waffle slab construction allows n considcr.iblc dead load reduction compared 10 conventional llat slab construction since the H•lumc of concrete used can be minimiied due 10 the short span between 1hc joiSls.

CHAPTER 12

586

Warne slabs arc particularly advantageous when the use or long span and/or loads is desired without the use or deepened drop panels, capnals or support beams he.,! cal applica1ions include parking garages or warehouses. A park mg garage wi1h a warn lyp1. system is shown in Figure 12.Jb. . . . . cslai, High form work and labour cosls arc the_ main re~sons !or limned applications of" nc slabs in contemporary concrete constructmn practice. af.

12.2.5 Slabs with Beams In s/a/J wirh beam.< system a solid slab panel is_ supported by beams_ on all four sides. \\iih this system, when the ra1io or 1hc sp~n lengths ,s two or more, load 1s predominately trans. [erred by bending in the short d1rec11on and_ 1_he panel csscn11ally acts as a one-way slah..\s the panel approaches a square shape, a s1gnihcant load 1s transrerrcd by bending in both or. thogonal directions, and the panel should be t'.eatcd as a two-"'.ay slab. Plan views and'"· tical sections for slab with beams arc shown m Frgurc 12.4a. rhc presence or beams m" require a greater storey height. A typical application of' the slab with beams system (par\. ing garage) is shown in Figure 12.4b.

12.3 BEHAVIOUR OF TWO-WAY SLABS SUBJECTED TO FLEXURE 112.3.1

I

Background Before we proceed with explaining different design approaches for two-way slabs, it is important for the reader to understand how these sys1ems behave under gravity loads. Flexural behaviour or two-way slabs is signilicantly inOuenccd by the support conditions. The following lhree cases will he considered: I)

Flat plate/slab panels supported only by columns at the comers; note the supponssho,n with arrows (see Figure 12.Sa). 2) Slab panels supported either by stiff beams or walls (slabs on stiff supports) hal'C con· tinuous supports along lhe edges, as shown in Figure 12.Sb; note lixed supponsonall sides. 3) Slab with beams on all sides - an intcnncdiate case between I) and 2), where a slab is su~ ported by columns (shown with arrows) and beams (shown as spring suppons), a< ,1. Justratcd in Figure 12.5c. It is important IO discuss the inOucncc or support conditions upon the magni1ude ofdcflli· tions in two-way slabs for the 1hrcc above described cases. The wtal slob deflection al nuJ span is the sum of lhe slab dcOcction at the edges plus the relative slob dellection bet•"' the edges and lhc midspan. Figure 12.5a shows O llat plate supported only by fourcolunin~ The midspan dcllcctions arc largest in this case since 1hc dcllections along 1he unre•trarnth< edges be1wccn the columns will cause an increase in 1he slab deOection 01 midspan. On ,JI 01her hand, dencctions in lhe slab supported along all four edges by sriff bewns or_ '~ 11 (Figure 12.SbJ arc smallcs1 since the dcOcc1ion is equal to zero along 1hc edges. f,n ;; 1 when a slab is supported by beams on all four sides (Figure 12.5c), 1he dellecnon d. midspan regions fall in between the other two cases because the beams arc cxpo:cted 10 ' • etl;haf"' Oect, but less th_an in lhc case of a lla1 plate. A conceptual diagram showing dellccl for lhc above discussed cases is presented in Figure J2.5d. Jen/LI" Con s rdcr Lwo slabs charac1crizcd by the same 1hickncss (180 mm) and span ,JJ!,r· 1 (5 m sq~areJ rhal arc subjected to the same uniform gravity load of 9.5 kPa. The onl) " ,,1~ encc rs rn rhc suppon cond11rons: one of the slabs is supported by columns at the come_ J,;· (Oat slab), while anolhcr is conrinuously supponcd al the edges by walls. In the fir;l_cast;tli'< placcment.s arc res1.raincd by the columns (which act like sticks with pinned connccuons

DESIGN OF TWO ·WAY SLABS

I,

587

re 12.s Two-way slabs with ~rent support conditions: flat slab/plate; b) slab on stiff pports; c) slab with beams on sides, and d) deflected shapes ,two-way slabs. a) Support (column) at each comer

Stiff beam or wall

b)

Support (column)

c)

,_~'r'''-"

d)

Slab on 1iff supports

Flat plate/slab (Case 1) (Case 2) top, similar 10 simple supports in beams), lhen:by allowing di,placemcnL, lllld ro1otions ut the four slab edges. In the lancr ca. holh for di,placcmcnL, lllld rotations and net like fixed supports. The« arc two extreme support conditions: the most Oe.,iblc und the most rigid. 1bc displacement contolJe; (regions with s-qWII displac Lhere is a significant drop in moment values just beyond. Moment rcdisuibuuon was a~ .10 . . -~~ served m the perpendicular span due 10 redistribution of bending moments tn . . 1in· span. Along Section 2-2 parallel wi1h column line AB (sec Figure 2.13c), a signiu,~,,n' crease in bending momcnls wa, observed in columns A and B rcla1ivc to the iowl m~ t,J1 (moment grou1cnt ... ,· JM ". Bending · moment at 1he column loca1ion was angina · · ·lly033~ · ed ·...t ii>' · 11··'.ncreascd lo 0.50M,, alter the onset of cracking. A reverse 1endcncy was Obscrv" • rtt'kin1· midspan · bending momcn1 decreased from 0.66 M,, Lo 0.47M,, after 1hc onSCI 01 c

se;;

DESIGN DF TW

D·WAY SLABS

597

a)

Load Level Service··············· Design ___ _

rtI

MULr = 2.65 M0cs

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Load Level Service ............. Design----

0

+1

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0

-1 c)

11 Moment redistribution in flat slabs: a) model tayout; b) transver5e moment distribution - Section • column A, and c) moment distribution in longitudinal direction - Section 2·2

::gt~re 12•13 (ad,

~

P ed from Rang an and Hale 1984 with the pefflli!Sion of the ""1trlcan concrete Institute).

CHAPTER 12

598

oncluded 1hal due lo signilicanl redistribution or bending m It can be C ' . ~ . orncnts · c designer has 10 ensure that the total flcxur,1! capacuy betwe '" 1,, way sIabs, lh · . .. .. en the , and midspan sections is equal to the total moment M.,. Tlus 1s one or ihc key rule 01 u~, lO the design or continuous flexural members and two-way slabs and will bed' s tclm~ . n 12 6 The actual distribution or bending moments and rcinl'orccmcnt .:"'uss,din I S&lO .. . . . . ~~ column and midspan sections 1s a secondary cons1dcrallon. and II depends on d . n1~ . .. d . H . I cs1gn po, liccs and standards followed ma spcc1\1c cs1gn. cncc, w 1en lota\ momcm M f •· 1 cilic s!ah span is given, the designer can select a number or different rcinl'orccmc;i°' • 1>1ror midspan and column regions which would result in lhc same ultimate load c amoun1; · ' apacnyro, 1he span.

12.3.5

Design for Flexure According to CSA A23.3 According to CSA A23.3 ~Ll3.5.I, a t'.vo-'."ay slab syslem can he designed using any pr, ccdure which satisfies cond111ons or cqu1hbnum and compal1h1!11y, provided 1hat ihe sirenm11 and serviceability requirements have been met The following design procedures arc omli;~ in CSA A 23.3 (Clauses 13.6 10 13.9): Direct Design Mc1hod Elastic Frame Method (also known as the Equivalent Frame Method) Elastic Plalc Theory, and Theorems or Plaslicily (e.g. the Yield Linc Method). The design or two-way slab syslcms must be performed considering both gral'ity and lateral loads. Some or the CSA A23.3 design procedures, like the Direct Design Mclhod.w, be used for gravity load analysis only, while others can he used both for gravity and lruml load analysis. Common design procedures will he explained in detail and illuslratedbydesign examples later in this chapter.

I 12.41 DEFINITIONS 12.4.1

Design Strip

strip is the ponion or a slab system bound laterally by the centrelines of the pan· IA23.3 Cl.13.1.2 I elsA de,·ign on each side or the column (CSA A23.3 Cl.13.9.2.1 ). This concept is illustrated'" Fig· ure 12.14, which shows a partial plan or a two-way floor slab without beams (flat plaie) ..\ typical slab panel is enclosed by [our columns (e.g. panel ABEF shown in Figure ll. 143 d A ponion or the slab al each floor level may be modelled as an equivalent beam, " '" width is reJ'crrcd lo as a design slrip. Equivalent beams al each floor level along wilh ih< supporting columns constitute a frame. In a general design scenario, several frame•_'; both directions, east-west (E-W) and north-south (N-S), need 10 he considered. Atypi~ frame ABC in the E-W direction is shown in Figure 12.14a, and the frame DBf'" ' N-S direction is shown in Figure 12.14b. The design strips in each direction aresho•• hatched in lhe ligurc. -, According 10 1hc CSA A23.3 notation, the longitudinal direction or the f~~ 4 referred lO as direction I, and the span in this direction is denoted as/ (see Figure !Z.1 _· Jctions compared to f1at plates. For that rca.~n. the minimum slab thickness. h,. is somewhat reduced and it can be dctennined from the following equation

A23.3 Eq. I3.2

JO I, I h, >_ /,(O.~+L~~J-(2.r')t.,

[12.6]

'"·here t, = additional thickness of the drop panel underneath the slah. and = drop panel overhang (dimension from the face of the column Ill the edge of drop panel).

X:

, mall ·r of the values determined in the two directions should be used for the slab thickThc S C . . . . p 0)) ncss calculation. that IS, .,, = mrn(.r" •.,.,,) (sec Figure -·- · The following dimensional limits should be met t.,Sh,

and -'J $

l,/4

.

I

Note that the maiumum -', va ue re

. I suits in a decrease in slab depth by approximate y

45 mm. . f b thickness Additional requirement or sIa , tes (see Figure I 2.2-). related to nat PIn

US

[ A23.3 Cl.l 3.2.5

01

discontinuous edges is the same as that

Orts Toe minimum thickness for slab_ panels

with Beams Between A uppcl as t ratio and the relative stiffness al bean1s J Slabs "th t,eams on all stdcsdepends on_~ pan h :::,labs with beams between nil supports is w1

.

.

in [WO dtn.'CttOOS-

The minimum tluc ncss, ,.

equaJ to [ A23.3 Eq.

13.3]

h, 2:

[ 12.7]

~! 30 +4/Ja.

I d ·hort direcuons; for cxamp e.

where

.

/J = rnuo o

p = I ' /1,. \or n panel

f Jear spans in Jong nn s

c and shown in Figlll1! 12.21,

az.¥-4-41

606

Figure 12.23

CHAPTER 12

Drop panel dimensions.

~r Drop panel Elevation

Drop panel

Column

.-1----

x" = min (x",x,,) X,12

Plan

am= average beam-to-slab stiffness ratio. Note that 1.0 $

p

$ 2.0, otherwise the slab should be treated as a one-way slab.

The beam-lo-slab stiffness ratio, a, is an average value obtained considering all beams along the panel edges, and it can be deier~ined using the procedures outlined in Section 12.4.5. For the given h, value, the required am value can be determined from the following equation

a = 1 [/,(0.6+ f,/1000) _ 30 ] "

4/3

h,

Relative stiffness of beams in the two orthogonal directions is an important paromc_1" influencing the deflections in two-way slabs. For example, when beams in one d,rcc_i;°" are signilicantly stiffer, the slab tends to act as a one-way slab spanning between th'~;/; beams (even 1f columns arc located on an essentially square grid). Nole that CSA • gives the following upper bound value for am: am$2.0

Minimum slab thicknesses for various slab types and different values of the key panull"'~ Panda;, arc summarized in Table 12.1.

Table 12.1

Minimum Thickness for Two-Way Slab Systems (Grade ~00 Reinfo~-

a.

p

Mlnlmum ~· ..

Flat plalc wilh edge beams

52.0

FJat pla1c whhout edge beams

52.0

-_JJ:~-/ f,7 ...

Two-way Slab System

1.0 Slab wilh beams between all supports 52.0

1.0 2,0

1.0 2.0

-- {/34_ ...

l,,/J!,__.-

-·n3s

/ /46

~

.

............al

DESIGN DF TWD-WAV SLABS

607

~ DESIGN DESIGN FOR FLEXURE ACCORDING TO THE DIRECT METHOD I A23.3 CLI l9 I This section describes the underlying concepts and design provisions for the Direct Design Method (DDM), a statics-based method for the design of two-way slabs adopted by CSA AZ3.3examples. and other codes. The application of this method will be demonstrated through two design

12.6.1

Limitations

I A23.3 Cl.3.9. l

CSA A23J prescribes that the DDM can be used when the following rcquiremenLs have been met (sec Figure 12.24): #I A slab must be regular (refer to Section 12.5.1 for more de1ails on regular two-way slabs). #2 There must be at least three continuous spans in each direction. #3 The successive span lengths (centre-to-centre or support-;) in each direction mus1 not differ by more than one-third of the longer span. _ #4 The DDM can be used only for gravity load analysis; gravity loads mus, be uniformly distributed over the entire slab panel. #5 The factored live load muSJ not exceed two times the factored dead load.

> Three spans (#2)

figure 12.24

Limitations of the Dilect Design Me th od. .. be used onlv for gravity load analysis_- The de. nt #4 states that the DDM can . for the slab ot a specific noor Requ,reme . I frame that consists of the design. I' 25 (the concept or design considers a part~:ve and below that level, as_shown i,";; used to design slob at level and colum~s~uced in Section !2A. I).~ s~m;lar is prevented by u roller support_ at sign stnp was m r N that lateral swaying ol t e ramc . on frame models for grovuy the top noor level. ote I to Section 12.7.Hor discusSJon the far end of the slab. Re er load analysis.

;tri f ' ;,,i~f~::

~

12.6.2 [ A23.3 Cl.I 3.9.2

~:~d

...

:~~~~;~n

t ,n~OOM~be~ . moments in • two-way sla~ the longuudmal dtre~The d1stnbulion of belnd1;::l us fir.a explain moment:~:" in Figure 12.26a The DDM ~s d b on examp e. not plate system s with a width equal to t c pla~n~o:Sider the span~ i°:n~ treats the slab as o ~~:h needs to be transfonne~ on plane lrambe su~jected to uniform urea li~th ; \ os shown ,n Figure I2.26b.. . stnp, The sla JS(baSCd on the de51gn strip w ,, design into t,ncar load w . 1,,

The Concep J

?'

:d~

IIJl@A,'i@ &JJI

CHAPTER 12

608

Figure 12,25

A gravity frame

model for the DOM.

conceptual diagram illustrating moment distribution f'or span AB can he seen in Figure 12.)(>: End moments M" and M" arc negative due to the restraints (columns) at points A and 8: while the midspan moment Mc is positive. Momeni distrihmion is similar to lhai for continuous beams and slabs discussed in Section 10.2.2. According to DDM, magnitudc1ei bending moments at points A, B, and C depend on several ractors, including the end pon conditions and the type of slab system (slab on beams or Oat plate/slab). Howmui, magnitude of moment gradient M0 is always equal to the sum or average value for lx!ndinf moments al the supporls A and Band the moment at the midspan C, as shown bclm,

,u,

However, M is also equal to the maximum moment or an equivalent simply support.Ii beam AB with th~ span I., subjected to uniform load w x 120 , that is,

Note that bending moments arc determined based on the clear span In (instead of 1hc ~~ntrc-to-ccntrc span /1). This is similar lO the design approach for continuous beams and slJ~ presented in Chapter I0. The above slatcmcnt can he proven by considering a frcc-hoi>~r-

i-(0.46 to 0.59)M 0

b)

t-0.26M 0 Figure 12,30 Moment distribucion at columns: a) an interior column, and b) an exterior column.

(-0.65M,) (see Table 12.2). One-third of that moment is assigned 10 band b centered at the column. The moment for the column strip is equal to -(0.46 10 0.59)1\,I . Toe remaining bending moment, equal to the difference between the column strip mo'~ent and the moment at band b,. should be resisted by the remaining portion of the column strip b)

outside band b,. Exterior columns: according to Cl.13. I0.3, the total factored negu1ive moment at an exterior column (M,) equal 10---0.26 M,should be resisted by the band b,. This is illu.,

di.stance of 1hc lirsl smd from the column face fs) stud rail .... pacing (g)

ChL·cJ... lhc

!\\

'

u-\\;1y shear rc,i:,,t:mcc for sl:1hs with 1",cam, l~L'C Section \ 2.9.5i.

IU

When a: (;_/l 1 ~ I -.0· beams provide 1hc entire shear resistance for the floor system (rctcr to Scc1wn 6.7 for shear design of beams). When O < a: l: '/: < 0. shc::ir is n.:sistcd hoLh hy the ht·:i.rn~ :md the .,l,1h (usc linL·ar i11tcrpt1lati(1n 10 finJ ihl' LIL'.,ign sh\:':1r i"nrcl', and ~trc:-~csl

Example 12.8 Two-Way Fial Plate • Shear Design

Consider o pion view of u two-WU)" not plote noor system designed in Exumple 12.1. shown in the skclch below. Use the slub thickness or IMO mm und the clTectivc depth or 1~0 mm. The foctorcd ureu loud is w1= 12.6 kPu. Dr.rig11 tlrt' sh1hj(1rsht'aracnmli11g ro the CSA r\23.3 Tf