100712 Design Slabs Shear_ACI 318

8/11/2019 100712 Design Slabs Shear_ACI 318

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Reinforced Concrete Designin Compliance with ACI 318M-08

Part 8: Design of Two-Way Slabs Shear

by

Dr. S. K. Ghosh

S. K. Ghosh Associates Inc.Palatine, IL and Aliso Viejo, CA

USA

A web seminar series in cooperation with and under sponsorship of theDepartment of Municipal Affairs, Emirate of Abu Dhabi, UAE

www.skghoshassociates.com

334 East Colfax Street, Unit E 43 Vantis DrivePalatine, IL 60067 Aliso Viejo, CA 92656Ph: (847) 991-2700 Fax: (847) 991-2702 Ph: (949) 249-3739 Fax: (949) 249-3989


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S. K. Ghosh Associates Inc.


- 1 -

REINFORCED CONCRETE DESIGN IN COMPLIANCE WITH

ACI 318M-08: DESIGN OF TWO-WAY SLABS - SHEAR

Please stay tuned. We will be starting at

8:00 am UAE Time.

You will be able to listen to the seminar

using computer speakers

If you are encountering technical difficulties, please call 00 1 847 991 2700

Visit us at: www.skghoshassociates.com

SKGA Web Seminarin cooperation with and under sponsorship of

the Department of Municipal Affairs, Emirate of Abu Dhabi, UAE

- 2 -

REINFORCED CONCRETE DESIGN IN

COMPLIANCE WITH ACI 318M-08

A web seminar series in cooperation wi th and under sponsorship of

the Department of Municipal Affairs, Emirate of Abu Dhabi, UAE.

Part 9: DESIGN OF TWO-WAY SLABS -

SHEAR


Palatine, IL and Aliso Viejo, CA



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- 3 -

Shear in Two-Way Slab Systems

Shear is typically not critical where a two-way slab

system is supported on beams and/or walls.

However, shear can be critical for flat plates or flat

slabs directly supported on columns. Shear strength

at an exterior slab-column connection is especially

critical because the total exterior negative slab

moment must be transferred to the edge column.

- 4 -

Unbalanced moment

Moment distribution: gravity load

Moment distribution: lateral load


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- 5 -


Two types of shear are checked for flat plates or flat

slabs directly supported by columns.

- One-way shear (beam-type shear)

- Two-way shear (punching shear)

- 6 -


One-way shear (beam-type shear)

- slab acts as a wide beam spanning between

columns

- critical section is taken at a distance d (effectivedepth of slab) away from the column face


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- 7 -


Two-way shear (punching shear)

- failure line occurs along the surface of a truncated

cone or pyramid around a column

- critical section is taken at a distance d/2 from the

face of the column

- two-way shear is usually more critical than one-way

shear

- 8 -



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- 9 -


Direct shear at an interior slab-column joint

(Source: Portland Cement Association, Notes on ACI 318-08 Building Code

Requirements for Structural Concrete, Skokie, IL, 2008)

One-way shear Two-way shear

- 1 0 -


When no or insignificant moment is transferred from

slab to column, direct shear, which is distributed

uniformly around the critical shear perimeter bo,

occurs around slab-column joints.

When significant moment is transferred from slab to

column because of unbalanced gravity loads on

either side of an interior column or horizontal load,

shear transferred from unbalanced moment, in

addition to direct shear, should be considered.


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- 1 1 -


ACI 318M-08 Fig. R11.11.7.2: Assumed distribution of shear stress

- 1 2 -

Crit ical Section for Shear

ACI 11.11 Provis ions for slabs and footings

ACI 11.11.1

The shear strength of slabs and footings in the vicinity

of columns, concentrated loads, or reactions is

governed by the more severe of two conditions

ACI 11.11.1.1. Beam action where each critical

section to be investigated extends in a plane across

the entire width.


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- 1 3 -


ACI 11.11.1.2. For two-way action, each of the critical

sections to be investigated shall be located so that its

perimeter bo is a minimum but need not approach closer

than d/2 to:

(a) Edges or corners of columns, concentrated loads, or

reaction areas; and

(b) Changes in slab thickness such as edges of capitals,

drop panels, or shear caps.

For two-way action, the slab or footing shall be designed

in accordance with 11.11.2 through 11.11.6

- 1 4 -


Tributary areas and critical sections for two-way shear

(Source: Portland Cement Association, Notes on ACI 318-08 Building Code


(Drop panel)


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- 1 9 -

Concrete Shear Strength for

Two-Way Act ion

ACI 11.11.2

The design of a slab or footing for two-way action is

based on Eq. (11-1) and (11-2). Vc shall be

computed in accordance with 11.11.2.1, 11.11.2.2,

or 11.11.3.1. Vs shall be computed in accordance

with 11.11.3. For slabs with shearheads, Vn shall be

in accordance with 11.11.4. When moment is

transferred between a slab and a column, 11.11.6

shall apply.

- 2 0 -


Two-Way Act ion

ACI 11.11.2.1. For nonprestressed slabs and footings,

Vc shall be the smallest of (a), (b), and (c):

(a)

Where is the ratio of long side to short side of thecolumn, concentrated load or reaction area;

dbf2

117.0V o'

cc

+= ACI Eq. (11-31)


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- 2 1 -


Two-Way Act ion

(b)

where s is 40 for interior columns, 30 for edgecolumns, 20 for corner columns; and

(c)

ACI Eq. (11-32)

ACI Eq. (11-33)

dbf2b

d083.0V o

'c

o

sc

+

=

dbf33.0V o'

cc =

- 2 2 -


Two-Way Act ion

ACI 318M-08 Fig. R11.11.2: Value of for a nonrectangular loaded area


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- 2 5 -

Shear Strength Provided by Bars, Wires,

and Single or Multiple-Leg Stirrups

ACI 11.11.3

Shear reinforcement consisting of bars or wires and

single- or multiple-leg stirrups shall be permitted in

slabs and footings with d greater than or equal to

150 mm, but not less than 16 times the shear

reinforcement bar diameter. Shear reinforcement

shall be in accordance with 11.11.3.1 through

11.11.3.4.

- 2 6 -



ACI 318M-08 Fig. R11.11.3 (a) (c): Single- or multiple-leg stirrup-type slabshear reinforcement

(a) Single-leg stirrup or bar (b) Multiple-leg stirrup or bar

(c) Closed stirrups


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- 2 7 -



ACI 11.11.3.1. Vn shall be computed by Eq. (11-2),

where Vc shall not be taken greater than

and Vs shall be calculated in accordance with 11.4.

In Eq. (11-15), Av shall be taken as the cross-

sectional area of all legs of reinforcement on one

peripheral line that is geometrically similar to the

perimeter of the column section.

dbf17.0 o'

c

scn VVV +=

s

dfAV

ytv

s =

ACI Eq. (11-2)

ACI Eq. (11-15)

- 2 8 -



ACI 11.11.3.2. Vn shall not be taken greater than

ACI 11.11.3.3. The distance between the column face

and the first line of stirrup legs that surrounds the

column shall not exceed d/2. The spacing between

adjacent stirrup legs in the first line of shear

reinforcement shall not exceed 2d measured in a

direction parallel to the column face.

dbf5.0 o'

c


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- 2 9 -



The spacing between successive lines of shear

reinforcement that surround the column shall not

exceed d/2 measured in a direction perpendicular to

the column face.

ACI 11.11.3.4. Slab shear reinforcement shall satisfy

the anchorage requirements of 12.13 and shallengage the longitudinal flexural reinforcement in the

direction being considered.

- 3 0 -



Design and detailing criteria for slabs with s tirrups (interior column)(Source: Portland Cement Association, Notes on ACI 318-08 Building Code


dbf17.0V o'cu

dbf50.0

s/dfAdbf17.0V

o'c

ytvo'cu

+


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- 3 1 -



Design and detailing criteria for slabs with s tirrups(edge and corner column)

(Source: Portland Cement Association, Notes on ACI 318-08 Building CodeRequirements for Structural Concrete, Skokie, IL, 2008)

- 3 2 -

Shear Strength Provided by

Headed Shear Stud Reinforcement

ACI 11.11.5

Headed shear stud reinforcement, placed

perpendicular to the plane of a slab or footing, shall

be permitted in slabs and footings in accordancewith 11.11.5.1 through 11.11.5.4.


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- 3 3 -



The overall height of the shear stud assembly shall

not be less than the thickness of the member less

the sum of: (1) the concrete cover on the top flexural

reinforcement; (2) the concrete cover on the base

rail; and (3) one-half the bar diameter of the tension

flexural reinforcement.

- 3 4 -



Where flexural tension reinforcement is at the bottom

of the section, as in a footing, the overall height of

the shear stud assembly shall not be less than the

thickness of the member less the sum of: (1) the

concrete cover on the bottom flexural reinforcement;(2) the concrete cover on the head of the stud; and

(3) one-half the bar diameter of the bottom flexural

reinforcement.


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- 3 5 -



Commentary to 11.11.5

Headed shear stud reinforcement was introduced in

the 2008 Code. Using headed stud assemblies as

shear reinforcement in slabs and footings requires

specifying the stud shank diameter, the spacing of

the studs, and the height of the assemblies for the

particular applications.

- 3 6 -



Tests (Joint ACI-ASCE Committee 421) show that vertical

studs mechanically anchored as close as possible to the

top and bottom of slabs are effective in resisting

punching shear. The bounds of the overall specified

height achieve this objective while providing areasonable tolerance in specifying that height as shown

in Figure R7.7.5.

Joint ACI-ASCE Committee 421, Shear Reinforcement for

Slabs (ACI 421.1R-99) (Reapproved 2006), American

Concrete Institute, Farmington Hills, MI, 1999, 15 pp.


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- 3 7 -



ACI 318M-08 Fig. R7.7.5: Concrete cover requirements for headed shearstud reinforcement

- 3 8 -



ACI 318M-08 Fig. R7.7.5: Concrete cover requirements for headed shearstud reinforcement


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- 3 9 -



ACI 11.11.5.1. For the critical section defined in

11.11.1.2, Vn shall be computed using Eq. (11-2),

with Vc and Vn not exceeding and

, respectively.

dbf25.0 o'

c

dbf66.0 o'

c

- 4 0 -



Vs shall be calculated using Eq. (11-15) with Av equal

to the cross-sectional area of all the shear

reinforcement on one peripheral line that is

approximately parallel to the perimeter of the

column section, where s is the spacing of theperipheral lines of headed shear stud reinforcement.

Avfyt/(bos) shall not be less than'

cf.170

s

dfAV

ytv

s = ACI Eq. (11-15)


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- 4 1 -



ACI 11.11.5.2. The spacing between the column face

and the first peripheral line of shear reinforcement

shall not exceed d/2. The spacing between

peripheral lines of shear reinforcement,measured in

a direction perpendicular to any face of the column,

shall be constant.

- 4 2 -



For prestressed slabs or footings satisfying 11.11.2.2, this

spacing shall not exceed 0.75d; for all other slabs and

footings, the spacing shall be based on the value of the

shear stress due to factored shear force and unbalanced

moment at the critical section defined in 11.11.1.2, andshall not exceed:

(a) 0.75d where maximum shear stresses due to factored

loads are less than or equal to and

(b) 0.5d where maximum shear stresses due to factored

loads are greater than

'

cf5.0

'

cf5.0


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- 4 3 -



Commentary to 11.11.5.2

The specified spacing between peripheral lines of

shear reinforcement are justified by experiments

(Joint ACI-ASCE Committee 421). The clear

spacing between the heads of the studs should be

adequate to permit placing of the flexural

reinforcement.

- 4 4 -



ACI 11.11.5.3. The spacing between adjacent shear

reinforcement elements, measured on the perimeter

of the first peripheral line of shear reinforcement,

shall not exceed 2d.

ACI 11.11.5.4. Shear stress due to factored shear

force and moment shall not exceed at

the critical section located d/2 outside the outermost

peripheral line of shear reinforcement.

dbf17.0 o'

c


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- 4 7 -

QuestionandAnswerSession


- 4 8 -

Openings in Slabs

ACI 11.11.6 Openings in slabs

When openings in slabs are located at a distance less

than 10 times the slab thickness from a

concentrated load or reaction area, or whenopenings in flat slabs are located within column

strips as defined in Chapter 13, the critical slab

sections for shear defined in 11.11.1.2 and

11.11.4.7 shall be modified as follows:


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- 4 9 -

Openings in Slabs

ACI 11.11.6.1. For slabs without shearheads, that part

of the perimeter of the critical section that is

enclosed by straight lines projecting from the

centroid of the column, concentrated load, or

reaction area and tangent to the boundaries of the

openings shall be considered ineffective.

ACI 11.11.6.2. For slabs with shearheads, theineffective portion of the perimeter shall be one-half

of that defined in 11.11.6.1.

- 5 0 -

Openings in Slabs

Effect of openings in slabs on shear strength(Source: Portland Cement Association, Notes on ACI 318-08 Building Code



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- 5 1 -

Transfer of Moment in Slab-

Column Connections

ACI considers transfer of moment between a slab and

a column in 13.5.3 and 11.11.7.

Critical section for shear due to moment transfer is

d(effective depth)/2 from the face of the column,

which is the same critical section for direct two-way

shear.

- 5 2 -


Column Connections

The total shear stress due to direct shear and shear

caused by moment transfer is

where

Vu = direct shear at the critical section

Ac = area of critical section

J

cM

A

Vv uv

c

uu

+=


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- 5 3 -


Column Connections

v = factor used to determine the unbalanced momenttransferred by eccentricity of shear at slab-column

connections

Mu = unbalanced moment

c = distance from centroid of critical section to

face of section where stress is being computed

J = property of critical section analogous to polar

moment of inertia

- 5 4 -


Column Connections

Shear stress dist ribution due to moment-shear transfer at slab-columnconnections

(Source: Portland Cement Association, Notes on ACI 318-08 Building CodeRequirements for Structural Concrete, Skokie, IL, 2008)


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- 5 5 -


Column Connections

ACI 13.5.3

When gravity load, wind, earthquake, or other lateral

forces cause transfer of moment between slab and

column, a fraction of the unbalanced moment shall

be transferred by flexure in accordance with

13.5.3.2 through 13.5.3.4.

ACI 13.5.3.1. The fraction of unbalanced moment nottransferred by flexure shall be transferred by

eccentricity of shear in accordance with 11.11.7.

- 5 6 -


Column Connections

ACI 13.5.3.2.A fraction of the unbalanced moment

given by fMu shall be considered to be transferredby flexure within an effective slab width between

lines that are one and one-half slab or drop panel

thickness (1.5h) outside opposite faces of thecolumn or capital, where Mu is the factored moment

to be transferred and

( ) 21

321

1

bbf

+= ACI Eq. (13-1)


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- 5 9 -


Column Connections

(b) For unbalanced moments at interior support, and for

edge columns unbalanced moments about an axis

perpendicular to the edge, increase f to as much as1.25 times the value from Eq. (13-1), but not more than

f= 1.0, provided that Vu at the support does not exceed0.4Vc. The net tensile strain t calculated for theeffective slab width defined in 13.5.3.2 (c2+3h) shall not

be less than 0.010.

The value of Vc in item (a) and (b) shall be calculated in

accordance with 11.11.2.1 (Eqs. 11-31, 11-32, 11-33).

- 6 0 -


Column Connections

Commentary to 13.5.3.3

At exterior supports, for unbalanced moments about an

axis parallel to the edge, the portion of moment

transferred by eccentricity of shear vMu may bereduced provided that the factored shear at the support(excluding the shear produced by moment transfer)

does not exceed 75 percent of the shear strength Vc asdefined in 11.12.2.1 for edge columns or 50 percent for

corner columns. Tests indicate that there is no

significant interaction between shear and unbalanced

moment at the exterior support in such cases.


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- 6 1 -


Column Connections

Evaluation of tests of interior supports indicate that

some flexibility in distributing unbalanced moments

transferred by shear and flexure is possible, but with

more severe limitations than for exterior supports.

For interior supports, the unbalanced moment

transferred by flexure is permitted to be increased

up to 25 percent provided that the factored shear

(excluding the shear caused by the momenttransfer) at the interior supports does not exceed 40

percent of the shear strength Vc as defined in11.11.2.1.

- 6 2 -


Column Connections

ACI 13.5.3.4. Concentration of reinforcement over the

column by closer spacing or additional

reinforcement shall be used to resist moment on the

effective slab width defined in 13.5.3.2.


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- 6 3 -

Transfer of moment in slab-

column connections

Often column strip reinforcement is concentrated near

the column to accommodate this unbalanced

moment. Available test data (Hanson and Hanson

1968) seem to indicate that this practice does not

increase shear strength but may be desirable to

increase the stiffness of the slab-column junction.

Hanson, N.W., and Hanson J.M., Shear and MomentTransfer between Concrete Slabs and Columns,

Journal, PCA Research and Development

Labortories, V.10, No.1, Jan. 1968, pp. 2-16.

- 6 4 -


column connections

ACI 11.11.7 Transfer of moment in slab-column

connections

ACI 11.11.7.1. Where gravity load, wind, earthquake,or other lateral forces cause transfer of unbalanced

moment Mu between a slab and column, fMu shallbe transferred by flexure in accordance with 13.5.3.


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- 6 5 -


column connections

The remainder of the unbalanced moment, vMu, shallbe considered to be transferred by eccentricity of

shear about the centroid of the critical section

defined in 11.11.1.2 where

v = (1 f) ACI Eq. (11-37)

- 6 6 -


column connections

Graphical solut ion of ACI Eqs. (13-1) and (11-37)(Source: Portland Cement Association, Notes on ACI 318-08 Building Code



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- 6 9 -


column connections

The shear stress due to factored shear force and

moment shall not exceed at the critical

section located d/2 outside the outermost line of

stirrup legs that surround the column.

)f.( 'c 170

- 7 0 -


column connections

ACI 318M-08 Fig. R11.11.7.2: Assumed distribution of shear stress


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- 7 3 -


column connections

Ac = area of concrete of assumed critical section

= 2d (c1 + c2 + 2d)

Jc = property of assumed critical section analogous to

polar moment of inertia

=

Similar equations may be developed for Ac and Jc, for

columns located at the edge or corner of a slab.

( ) ( ) ( )( )2

dcdcd

6

ddc

6

dcd2

123

1

3

1 ++++

++

- 7 4 -


column connections

Section properties for shear stress computations(Source: Portland Cement Association, Notes on ACI 318-08 Building Code



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- 7 5 -


column connections



- 7 6 -


column connections




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- 7 7 -


column connections



- 7 8 -

Factored Shear in Slab Systems

with Beams

ACI 13.6.8 Factored shear in slab systems wi th

beams

ACI 13.6.8.1. Beams with f1l2/l1 equal to or greaterthan 1.0 shall be proportioned to resist shear

caused by factored loads on tributary areas which

are bounded by 45-degree lines drawn from the

corners of the panels and the centerlines of the

adjacent panels parallel to the long sides.


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- 7 9 -


with Beams

f1 = [(EcbIb)/(EcsIs)] in direction l1l1 = length of span in direction that moments are

being determined

l2 = length of span in direction perpendicular to l1.

EcbIb = Flexural stiffness of beam

Ecs

Is

= Flexural stiffness of slab

- 8 0 -


with Beams

ACI 318M-08 Fig. R13.6.8: Tributary area for shear on an interior beam


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- 8 1 -


with Beams

ACI 13.6.8.2. In proportioning beams with f1l2/l1 lessthan 1.0 to resist shear, linear interpolation,

assuming beams carry no load at f1 = 0, shall bepermitted.

ACI 13.6.8.3. In addition to shears calculated

according to 13.6.8.1 and 13.6.8.2, beams shall be

proportioned to resist shears caused by factoredloads applied directly on beams.

- 8 2 -


with Beams

ACI 13.6.8.4. Computation of slab shear strength on

the assumption that load is distributed to supporting

beams in accordance with 13.6.8.1 or 13.6.8.2 shall

be permitted. Resistance to total shear occurring on

a panel shall be provided.

ACI 13.6.8.5. Shear strength shall satisfy the

requirements of Chapter 11.


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- 8 3 -

BREAK!


- 8 4 -

Question

and

Answer

Session



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- 8 5 -

Example 1: Two-Way Slab Shear

Design

Check shear for the shaded design strip as shownbelow

5.5 m

5.5 m

5.5 m

4.3 m 4.3 m 4.3 m

Design Direction

- 8 6 -


Design

Structural system

Two-way slab systems without beams (flat plate

slab system); no edge beams

Material properties

Concrete: fc = 40 MPa wc = 2400 kg/m3

Reinforcement: fy = 420 MPa


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- 8 7 -


Design

Member dimension

Column: 400 mm 400 mm

Loading condition

Live load = 2.0 kN/m2

Partition load = 1.0 kN/m2

- 8 8 -


Design

Determine slab thickness

From ACI Table 9-5 (c)

h = ln / 30

= (5500 400) / 30 = 170 mm

Use 200 mm

According to ACI 9.5.3.2, allowable minimum

thickness is 125 mm < 200 mm O.K.


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- 8 9 -


Design

Effective depth d = 200 32 (20 mm cover and

diameter of one No.12 bar = 20 +12 = 32 mm) =

168 mm

- 9 0 -


Design

Determine factored load quDead load (qD):

qD = 0.2 2400 9.8/1000 = 4.7 kN/m2

Live load (qL):

Reducible live load (2009 IBC Section 1607.9.2)

R = 0.861(A 13.94)

= 0.861(23.65 13.94) = 8.4 %

where A = 4.3 5.5 = 23.65 m2


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- 9 1 -


Design

Reducible live load = (1-0.084)2.0 = 1.83 kN/m2

Non-reducible live load (Partition) = 1.0 kN/m2

qL = 1.83 + 1.0 = 2.83 kN/m2

qu = 1.4qD = 1.44.7 = 6.58 kN/m2

= 1.2qD+1.6qL = 1.24.7+1.62.83= 10.2 kN/m2 (Governs)

- 9 2 -


Design

Check one-way shear (interior column)

Panel centerline

4.3 m

5.5 m

Critical section

for one-way shear

0.4 m

0.4 m

Slab effective depth

d= 0.168 m

Design

Direction


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- 9 3 -


Design

Factored shear at critical section:

Vu = 10.2(5.5/2 0.4/2 0.168)4.3

= 104 kN

Vc= 0.750.17(fc)0.5bwd

= 0.750.171.0(40)0.543001681/1000

= 583 kN > Vu (= 104 kN) O.K.

- 9 4 -


Design

Check one-way shear (edge column)

Panel

centerline

4.3 m

5.5/2 = 2.75 mCritical section

for one-way shear0.4 m

0.4 m

Slab effective depth

d= 0.168 m

Design Direction


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- 9 5 -


Design


Vu = 10.2(2.75 0.4/2 0.168)4.3

= 104 kN

Vc= 0.750.17(fc)0.5bwd

= 0.750.171.0(40)0.543001681/1000= 583 kN > Vu (= 104 kN) O.K.

- 9 6 -


Design

Check two-way shear (Interior column)

Panel centerline

4.3 m

5.5 m

Critical section

for two-way shear

0.4 m

0.4 m

d/2= 0.084 md/2= 0.084 mDesign

Direction


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- 9 7 -


Design

Side dimension of square critical section = 400 + 168

= 568 mm


Vu = 10.2(5.54.3 0.5680.568)

= 238 kN

- 9 8 -


Design

Shear strength Vc for two-way slab (ACI 11.11.2.1)

= 0.4 / 0.4 = 1.0

s = 40 (Interior)

bo = 4568 = 2272 mm

dbf2

117.0V o'

cc

+=

dbf2b

d

083.0V o'

co

s

c

+

=

dbf33.0V o'

cc =

ACI Eq. (11-31)

ACI Eq. (11-32)

ACI Eq. (11-33)


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- 9 9 -


Design

Vc = 0.75 797 = 598 kN > Vu (= 238 kN) O.K.

kN12311000/1682272401

2117.0

dbf2

117.0V o'

cc

=

+=

+=

kN9931000/16822724022272

16840083.0

dbf2b

d083.0V o

'

c

o

sc

=

+

=

+

=

kN7971000/16822724033.0dbf33.0V o'cc === (Governs)

- 100 -


Design

Check two-way shear (edge column)

Panel

centerline

4.3 m


for two-way shear0.4 m

0.4 m

d/2= 0.084 m

Design Direction

d/2= 0.084 m


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- 101 -


Design

Dimension of critical section, b1 = 400 + 168 = 568 mm

Dimension of critical section, b2 = 400 + 168/2 = 484 mm


Vu = 10.2[(5.5/2+0.2)4.3 0.5680.484]

= 127 kN

- 102 -


Design

Shear strength Vc for two-way slab (ACI 11.11.2.1)

= 0.4 / 0.4 = 1.0

s = 30 (Edge column)

bo = 568 + 484 + 484 = 1536 mm

dbf2

117.0V o'

cc

+=

dbf2b

d

083.0V o'

co

s

c

+

=

dbf33.0V o'

cc =

ACI Eq. (11-31)

ACI Eq. (11-32)

ACI Eq. (11-33)


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52



- 103 -


Design

Vc = 0.75 539 = 404 kN > Vu (= 127 kN) O.K.

Slab thickness is determined as 200 mm.

kN8321000/1681536401

2117.0

dbf2

117.0V o'

cc

=

+=

+=

kN7151000/16815364021536

16830083.0

dbf2b

d083.0V o

'

c

o

sc

=

+

=

+

=

kN5391000/16815364033.0dbf33.0V o'

cc === (Governs)

- 104 -


Design

Direct Design Method can be used for this example

(13.6.1). Positive and negative moments at column

and middle strips were determined using Direct

Design Method.


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53



- 105 -


Design

Total factored moment (Mo)

where

qu = factored load (10.2 kN/m2)l2 = length perpendicular to design direction (4.3m)

ln = clear span length in design direction (5.1m)

mkN6.1428

1.53.42.10

8

qM

2

2

n2uo

=

=

= ll

- 106 -


Design

Table: Moments and provided reinforcementLocation Mu (kN-m) Width

(mm)

Required

As (mm2)

Provided

reinforcement

End Span Column

strip

Ext. neg. 0.26Mo = 37.1 2150 583 9 No.12

Positive 0.31Mo = 44.2 2150 696 7 No.12

Int. neg. 0.53Mo =75.6 2150 1201 11 No.12

Middle strip Ext. neg. 0.0Mo = 0 2150 0 7 No.12

Positive 0.21Mo = 30.0 2150 471 7 No.12

Int. neg. 0.17Mo = 44.2 2150 696 7 No.12

Interior

Span

Column

strip

Positive 0.21Mo = 30.0 2150 471 7 No.12

Negative 0.49Mo = 69.9 2150 1109 11 No.12

Middle strip Positive 0.14Mo = 20.0 2150 313 7 No.12

Negative 0.16Mo = 22.8 2150 357 7 No.12

Minimum As = 0.0018bh = 0.00182150200 = 774 mm2, Maximum spacing = 2h = 400 mm, Use 7-No.12 (A s = 791 mm

2) to

satisfy minimum As and maximum spacing


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- 107 -


Design

Shear Design

Total shear stress is the sum of the direct shear stress

plus the shear stress due to the fraction of

unbalanced moment transferred by eccentricity of

shear. Assume shear stress due to moment transfer

by eccentricity of shear varies linearly about thecentroid of the section.

- 108 -


Design

Case 1: Shear check at end support

Panel

centerline

4.3 m


for two-way shear0.4 m

0.4 m

d/2= 0.084 m

Design Direction

d/2= 0.084 m


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- 109 -


Design

Direct shear Vu can be computed as below

A = area tributary to column

= (5.5/2+0.2)4.3 = 12.69 m2

b1 = dimension of critical section in direction

of analysis = 0.4 + 0.168/2 = 0.484 m

( )n

2121uu

MMbbAqV

l

=

- 110 -


Design

b2 = dimension of critical section perpendicular to

direction of analysis

= 0.4 + 0.168 = 0.568 m

M1 = total negative design strip moment at interiorsupport determined from Direct Design Method

= 0.70Mo = 99.8 kN-m

M2 = total negative design strip moment at exterior

support determined from Direct Design Method

= 0.26Mo = 37.1 kN-m


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57



- 113 -


Design

0.4Vc (= 0.4404 = 162 kN) > Vu (= 114.3 kN)

Use 5-No.12 bars over effective width equal to c2 + 3h = 400

+ 3200 = 1000 mm. From section analysis, a (the depthof rectangular stress block) is 7.0 mm.

c (distance from top to neutral axis) = a/1= 7/0.76 = 9.2 mm

t (net tensile strain) = 0.003(d-c)/c= 0.003(168-9.2)/9.2 = 0.052 > 0.01

Thus, f= 1.250.62 = 0.78; v = 1-f= 1 0.78 = 0.22

Mu = 0.3Mo = 42.8 kN-m (ACI 13.6.3.6)

- 114 -


Design

Therefore,

( ) ( )

( ) ( )

3

32

1

213

2121

m0464.0

484.06

568.0484.02168.0568.02484.0168.0484.02

b6

bb2db2bdb2

c

J

=

+++

=

+++=

2

uv

c

uu

m/kN6430464.0

8.4222.0

26.0

3.114

J

cM

A

Vv

=

+=

+=


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- 115 -


Design

Shear strength

vc = 0.750.33(40)0.5 = 1.56 N/mm2 = 1560 kN/m2

> Vu (= 643 kN/m2) O.K.

- 116 -


Design

Case 2: Shear check at interior support

Panel centerline

4.3 m

5.5 m

Critical section

for two-way shear

0.4 m

0.4 m

d/2= 0.084 md/2= 0.084 mDesign

Direction


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59



- 117 -


Design

where

A = 5.54.3 = 23.65 m2

b1 = b2 = 0.4 + 0.168 = 0.568 m

M1 = 0.70Mo = 99.8 kN-m

M2 = 0.26Mo = 37.1 kN-m

( )

[ ] kN3.2503.120.2381.5

1.378.99568.065.232.10

MMbbAqV

2

n

2121uu

=+=

+=

+=

l

- 118 -


Design

Total shear stress is computed.

Ac = bod = 2(0.568+0.568)0.168

= 0.38 m2

( )

( ) 33

3

211

m0738.03

168.0568.03568.0168.0568.0

3

db3bdb

c

J

=++

=

++=


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60



- 119 -


Design

The difference between the slab moments acting on

opposite faces of the interior support needs to be

transferred by shear to the first interior column.

Exterior moment at the face of the support is 0.70Mo =

0.70142.6 = 99.8 kN-m, and the interior moment atthe face of the support is 0.65Mo = 0.65142.6 =92.7 kN-m. Therefore, the unbalanced moment M

u

=

99.8 92.7 = 7.1 kN-m

- 120 -


Design

0.4Vc

(= 0.4598 = 239 kN) < Vu

(= 250.3 kN)

Thus, there is no increase in f(= 0.6).

v = 1 0.6 = 0.4

( )60.0

568.0568.0321

1f =

+=


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61



- 121 -


Design

vc = 0.750.33(40)0.5 = 1.56 N/mm2 = 1560 kN/m2

> Vu (= 697 kN/m

2

) O.K.

2

uv

c

uu

m/kN6970738.0

1.740.0

38.0

3.250

J

cM

A

Vv

=

+=

+=

- 122 -

Question

and

Answer

Session



64/64


- 123 -

Thank You!!

For more information


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100712 Design Slabs Shear_ACI 318

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Transcript of 100712 Design Slabs Shear_ACI 318