By Dr. Attaullah Shah Swedish College of Engineering and Technology Wah Cantt. Reinforced Concrete...

By

Dr. Attaullah Shah

Swedish College of Engineering and Technology Wah Cantt.

Reinforced Concrete Design-8

Design of 2 way Slabs

One way vs Two way slab system

As ≥ Temp. steelMin. Spacing ≥ main steel ≥ 4/3 max agg. ≥ 2.5 cm (1in) ∅Max. Spacing ≤ 3 t ≤ 45 cm (17in)

Min slab thickness =

L>2S

Shear Strength of SlabsShear Strength of Slabs

In two-way floor systems, the slab must have adequate thickness to resist both bending moments and shear forces at critical section. There are three cases to look at for shear.

Two-way Slabs supported on beams

Two-Way Slabs without beams

Shear Reinforcement in two-way slabs without beams.

1.

2.

3.


Two-way slabs supported on beams

The critical location is found at d distance from the column, where

The supporting beams are stiff and are capable of transmitting floor loads to the columns.

bdfV 2 cc


The shear force is calculated using the triangular and trapezoidal areas. If no shear reinforcement is provided, the shear force at a distance d from the beam must equal

where,

bdfVV 2 ccud

d

lwV

2

2uud


There are two types of shear that need to be addressed

Two-Way Slabs without beams

One-way shear or beam shear at distance d from the column

Two-way or punch out shear which occurs along a truncated cone.

1.

2.


One-way shear or beam shear at distance d from the column

Two-way or punch out shear which occurs along a truncated cone.

1.

2.


One-way shear considers critical section a distance d from the column and the slab is considered as a wide beam spanning between supports.

bdfVV 2 ccud


Two-way shear fails along a a truncated cone or pyramid around the column. The critical section is located d/2 from the column face, column capital, or drop panel.


If shear reinforcement is not provided, the shear strength of concrete is the smaller of:

perimeter of the critical section ratio of long side of column to short side

bo = c =

dbfdbfV ococ

c

c 4 4

2


If shear reinforcement is not provided, the shear strength of concrete is the smaller of:

s is 40 for interior columns, 30 for edge columns, and 20 for corner columns.

dbfb

dV oc

o

sc 2


Shear Reinforcement in two-way slabs without beams.

For plates and flat slabs, which do not meet the condition for shear, one can either

- Increase slab thickness - Add reinforcement

Reinforcement can be done by shearheads, anchor bars, conventional stirrup cages and studded steel strips.


Shearhead consists of steel I-beams or channel welded into four cross arms to be placed in slab above a column. Does not apply to external columns due to lateral loads and torsion.


Anchor bars consists of steel reinforcement rods or bent bar reinforcement


Conventional stirrup cages


Studded steel strips


The reinforced slab follows section 11.12.4 in the ACI Code, where Vn can not

The spacing, s, can not exceed d/2.

If a shearhead reinforcement is provided

dbfVVV ocscn 6

s

dfAV

dbfV

yvs

occ 4

dbfV ocn 7

Example ProblemExample Problem

Determine the shear reinforcement required for an interior flat panel considering the following: Vu= 195k, slab thickness = 9 in., d = 7.5 in., fc = 3 ksi, fy= 60 ksi, and column is 20 x 20 in.


Compute the shear terms find b0 for

c c 0 4V f b d

0

column4 4 20 in. 7.5 in.

width

110 in.

b d


Compute the maximum allowable shear

Vu =195 k > 135.6 k Shear reinforcement is need!

c c 0 4

1 k0.75 4 3000 110 in. 7.5 in.

1000 lbs

135.6 k

V f b d


Compute the maximum allowable shear

So Vn >Vu Can use shear reinforcement

c c 0 6

1 k0.75 6 3000 110 in. 7.5 in.

1000 lbs

203.3 k

V f b d


Use a shear head or studs as in inexpensive spacing. Determine the a for

c c 0 2V f b d

0

column4 2

widthb a


Determine the a for

The depth = a+d

= 41.8 in. +7.5 in. = 49.3 in. 50 in.

u c 0 2

19500 lb 0.75 2 3000 4 20 in. 2 7.5 in.

41.8 in.

V f b d

a

a


Determine shear reinforcement

The Vs per side is Vs / 4 = 14.85 k

s u c

195 k 135.6 k

59.4 k

V V V


Determine shear reinforcement

Use a #3 stirrup Av = 2(0.11 in2) = 0.22 in2

s

14.85 k19.8 k

0.75V

v y v ys

s

A f d A f dV s

s V


Determine shear reinforcement spacing

Maximum allowable spacing is

7.5 in.3.75 in.

2 2

d

2v y

s

0.22 in 60 ksi 7.5 in.

19.8 k

5.0 in.

A f ds

V


Use s = 3.5 in.

The total distance is 15(3.5 in.)= 52.5 in.

50 in.# of stirrups 14.3 Use 15 stirrups

3.5 in.


The final result:

15 stirrups at total distance of 52.5 in. So that a = 45 in. and c = 20 in.

Direct Design Method for Two-way SlabDirect Design Method for Two-way Slab

Minimum of 3 continuous spans in each direction. (3 x 3 panel)

Rectangular panels with long span/short span 2

Method of dividing total static moment Mo into positive and negative moments.

Limitations on use of Direct Design method

1.

2.

Direct Design Method for Two-way SlabDirect Design Method for Two-way Slab


Successive span in each direction shall not differ by more than 1/3 the longer span.

3.

4. Columns may be offset from the basic rectangular grid of the building by up to 0.1 times the span parallel to the offset.

Direct Design Method for Two-way Direct Design Method for Two-way SlabSlabLimitations on use of Direct Design method

All loads must be due to gravity only (N/A to unbraced laterally loaded frames, from mats or pre-stressed slabs)

Service (unfactored) live load 2 service dead load

5.

6.

Direct Design Method for Two-way Direct Design Method for Two-way SlabSlab

For panels with beams between supports on all

sides, relative stiffness of the beams in the 2

perpendicular directions.

Shall not be less than 0.2 nor greater than 5.0


7.

212

221

l

l

Definition of Beam-to-Slab Stiffness Ratio, Definition of Beam-to-Slab Stiffness Ratio,

Accounts for stiffness effect of beams located along slab edge reduces deflections of panel

adjacent to beams.

slab of stiffness flexural

beam of stiffness flexural

Definition of Beam-to-Slab Stiffness Ratio, Definition of Beam-to-Slab Stiffness Ratio,

With width bounded laterally by centerline of adjacent panels on each side of the beam.

scs

bcb

scs

bcb

4E

4E

/4E

/4E

I

I

lI

lI

slab uncracked of inertia ofMoment I

beam uncracked of inertia ofMoment I

concrete slab of elasticity of Modulus E

concrete beam of elasticity of Modulus E

s

b

sb

cb

Two-Way Slab DesignTwo-Way Slab Design

Static Equilibrium of Two-Way Slabs

Analogy of two-way slab to plank and beam floor

Section A-A:

Moment per ft width in planks

Total Moment

ft/ft-k 8

21wlM

ft-k 8

21

2f

lwlM



Analogy of two-way slab to plank and beam floor

Uniform load on each beam

Moment in one beam (Sec: B-B)ft-k

82

221

lb

lwlM

k/ft 2

1wl



Total Moment in both beams

Full load was transferred east-west by the planks and then was transferred north-south by the beams;

The same is true for a two-way slab or any other floor system.

ft-k 8

22

1

lwlM

Basic Steps in Two-way Slab DesignBasic Steps in Two-way Slab Design

Choose layout and type of slab.

Choose slab thickness to control deflection. Also, check if thickness is adequate for shear.

Choose Design method− Equivalent Frame Method- use elastic frame analysis to compute

positive and negative moments− Direct Design Method - uses coefficients to compute positive and

negative slab moments

1.

2.

3.

Basic Steps in Two-way Slab Basic Steps in Two-way Slab DesignDesign

Calculate positive and negative moments in the slab.

Determine distribution of moments across the width of the slab. - Based on geometry and beam stiffness.

Assign a portion of moment to beams, if present.

Design reinforcement for moments from steps 5 and 6.

Check shear strengths at the columns

4.

5.

6.

7.

8.

Minimum Slab Thickness for two-way constructionMinimum Slab Thickness for two-way construction

Maximum Spacing of Reinforcement

At points of max. +/- M:

Min Reinforcement Requirements

7.12.3 ACI in. 18 and

13.3.2 ACI 2

s

ts

s min s T&S from ACI 7.12 ACI 13.3.1A A

Distribution of MomentsDistribution of MomentsSlab is considered to be a series of frames in two directions:

Distribution of MomentsDistribution of Moments

Slab is considered to be a series of frames in two directions:

Distribution of MomentsDistribution of Moments

Total static Moment, Mo

3-13 ACI 8

2n2u

0

llwM

cn

n

2

u

0.886d h using calc. columns,circular for

columnsbetween span clear

strip theof width e transvers

areaunit per load factored

l

l

l

wwhere

Column Strips and Middle StripsColumn Strips and Middle Strips

Moments vary across width of slab panel

Design moments are averaged over the width of column strips over the columns & middle strips between column strips.


Column strips Design w/width on either side of a column centerline

equal to smaller of

1

2

25.0

25.0

l

l

l1= length of span in direction moments are being determined.

l2= length of span transverse to l1


Middle strips: Design strip bounded by two column strips.

Positive and Negative Moments in PanelsPositive and Negative Moments in Panels

M0 is divided into + M and -M Rules given in ACI sec. 13.6.3

Moment DistributionMoment Distribution

Positive and Negative Moments in Positive and Negative Moments in PanelsPanels

M0 is divided into + M and -M Rules given in ACI sec. 13.6.3

8

2n2u

0avguu

llwMMM

Longitudinal Distribution of Moments Longitudinal Distribution of Moments in Slabsin Slabs

For a typical interior panel, the total static moment is divided into positive moment 0.35 Mo and negative moment of 0.65 Mo.

For an exterior panel, the total static moment is dependent on the type of reinforcement at the outside edge.

Distribution of MDistribution of M00


The factored components of the moment for the beam.

Transverse Distribution of MomentsTransverse Distribution of Moments

The longitudinal moment values mentioned are for the entire width of the equivalent building frame. The width of two half column strips and two half-middle stripes of adjacent panels.


Transverse distribution of the longitudinal moments to middle and column strips is a function of the ratio of length l2/l1,1, and t.


Transverse distribution of the longitudinal moments to middle and column strips is a function of the ratio of length l2/l1,1, and t.

torsional constant

3

63.01

2

3

scs

cbt

scs

bcb1

yx

y

xC

IE

CE

IE

IE

Distribution of MDistribution of M00

ACI Sec 13.6.3.4

For spans framing into a common support negative moment sections shall be designed to resist the larger of the 2 interior Mu’s

ACI Sec. 13.6.3.5

Edge beams or edges of slab shall be proportioned to resist in torsion their share of exterior negative factored moments

Factored Moment in Column Strip Factored Moment in Column Strip

Ratio of flexural stiffness of beam to stiffness of slab in direction l1.

Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length)

t

Factored Moment in an Interior Strip Factored Moment in an Interior Strip

Factored Moment in an Exterior PanelFactored Moment in an Exterior Panel

Factored Moment in Column Strip Factored Moment in Column Strip

Ratio of flexural stiffness of beam to stiffness of slab in direction l1.

Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length)

t

Factored MomentsFactored Moments

Factored Moments in beams (ACI Sec. 13.6.3)

Resist a percentage of column strip moment plus moments due to loads applied directly to beams.

Factored MomentsFactored Moments

Factored Moments in Middle strips (ACI Sec. 13.6.3)

The portion of the + Mu and - Mu not resisted by column strips shall be proportionately assigned to corresponding half middle strips.

Each middle strip shall be proportioned to resist the sum of the moments assigned to its 2 half middle strips.

ACI Provisions for Effects of ACI Provisions for Effects of Pattern LoadsPattern Loads

The maximum and minimum bending moments at the critical sections are obtained by placing the live load in specific patterns to produce the extreme values. Placing the live load on all spans will not produce either the maximum positive or negative bending moments.

ACI Provisions for Effects of ACI Provisions for Effects of Pattern LoadsPattern Loads

The ratio of live to dead load. A high ratio will increase the effect of pattern loadings.

The ratio of column to beam stiffness. A low ratio will increase the effect of pattern loadings.

Pattern loadings. Maximum positive moments within the spans are less affected by pattern loadings.

1.

2.

3.

Reinforcement Details LoadsReinforcement Details Loads

After all percentages of the static moments in the column and middle strip are determined, the steel reinforcement can be calculated for negative and positive moments in each strip.

2uysu

2 bdR

adfAM

Reinforcement Details LoadsReinforcement Details Loads

Calculate Ru and determine the steel ratio , where =0.9. As = bd. Calculate the minimum As from ACI codes. Figure 13.3.8 is used to determine the minimum development length of the bars.

c

yu

ucuu

59.01

f

fw

wfwR

Minimum extension for Minimum extension for reinforcement in slabs without reinforcement in slabs without beams(Fig. 13.3.8)beams(Fig. 13.3.8)


The factored components of the moment for the beam.


The longitudinal moment values mentioned are for the entire width of the equivalent building frame. The width of two half column strips and two half-middle stripes of adjacent panels.

Limitations of Direct Design Method

By Dr. Attaullah Shah Swedish College of Engineering and Technology Wah Cantt. Reinforced Concrete...

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