Palestine Polytechnic University Collage of Engineering and Technology Department of Civil and Architectural Engineering Engineering
Reinforced Concrete II Dr. Nasr Younis Abboushi
2013-2014
Reinforced Concrete II
Dr. Nasr Abboushi
CONTENTS CHAPTER 8
CHAPTER 9
ONE-WAY SLABS
185
8.1
Types of slabs
185
8.2
Analysis of continuous beams and frames
187
8.3
Analysis and design of one-way solid slabs. ACI code limitations.
191
8.4
Minimum reinforcement ratio
195
8.5
Temperature and shrinkage reinforcement reinforcement
196
8.6
Reinforcement details
197
8.7
One-way joist floors and one-way ribbed slabs
212
8.8
Design of one-way ribbed slab
216
TWO-WAY SLABS
234
9.1
Introduction
234
9.2
Types of two-way slabs
234
9.3
Economical choice of concrete floor systems
237
9.4
Minimum thickness of two-way slabs
238
9.5
Slab reinforcement reinforcement requirements
241
9.6
Shear strength of two-way slabs
243
9.6.1
Two-Way Slabs Supported on Beams
244
9.6.2
Two-Way Slabs Without Beams
244
9.6.3
Tributary Areas for Shear in Two-Way Slabs
247
9.6.4
Shear Reinforcement Reinforcement in Two-Way Slabs Without Beams
247
9.7
Analysis and design of two-way slabs
249
9.8
Slab analysis by the coefficient method
250
9.9
Slab analysis by the direct design method (DDM).
294
9.9.1
Limitations on the Use of the Direct-Design Method
294
9.9.2
Column and middle strips
295
9.9.3
Total Static Moment at Factored Loads
296
9.9.4
Assignment of positive and negative moments
296
9.9.5
Lateral Distribution of Moments (between Column Strips and Middle Strips)
298
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CHAPTER 10
CHAPTER 11
CHAPTER 12
Dr. Nasr Abboushi
9.10
Slab analysis by the equivalent frame method (EFM)
302
9.11
Shear design in flat plates
305
STAIRS
317
10.1
Introduction
317
10.2
Types of stairs
318
10.3
Slab type stairs. Structural system
327
FOOTINGS AND FOUNDATIONS
341
11.1
Introduction
341
11.2
Types of footings
342
11.3
Distribution of soil pressure. Gross and net soil pressures
343
11.4
Design considerations
346
11.4.1
Size of footings
346
11.4.2
One-way shear (Beam shear)
346
11.4.3
Two-way shear (Punching shear)
347
11.4.4
Flexural strength and footing reinforcement reinforcement
348
11.4.5
Transfer of Load from Column to Footing
350
11.4.6
Bearing Strength
351
11.5
Spread (isolated) footings
353
11.6
Strip (wall) footings
360
11.7
Footings under eccentric column loads
363
11.8
Combined footings
372
11.9
Continuous footings
380
11.10
Mat foundations
386
DEVELOPMENT, DEVELOPMENT, ANCHORAGE, AND SPLICING OF REINFORCEMENT
393
12.1
Introduction
393
12.2
Flexural bond
393
12.3
Mechanism of bond transfer
394
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12.4
Development length
396
12.5
Hooked anchorages
405
12.6
Bar cutoffs and development of bars in flexural members
412
12.7
Development of positive moment reinforcement reinforcement
417
12.8
Development of negative moment reinforcement
421
12.9
Reinforcement continuity and structural integrity requirements
425
12.10
Splices of reinforcement
446
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CHAPTER 8
Dr. Nasr Abboushi
ONE-WAY SLABS
8.1 TYPES OF SLABS
Structural concrete slabs are constructed to provide flat surfaces, usually horizontal, in building floors, roofs, bridges, and other types of structures. The slab may be supported by walls, by reinforced concrete beams usually cast monolithically with the slab, by structural steel beams, by columns, or by the ground. The depth of a slab is usually very small compared to its span.
Structural concrete slabs in buildings may be classified as follows: 1.
One-way stabs: If a slab is supported on two opposite opposite sides only, it will will bend or
deflect in a direction perpendicular to the supported edges. The structural action is one way, and the loads are carried by the slab in the deflected short direction. This type of slab is called a one-way slab. 185
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If the slab is supported on four sides and the ratio
of the long side to the short side is equal to or
(
), most of
the load (about
or more) is carried in the
greater than
short direction, and one-way action is considered for all practical purposes. If the slab is made of reinforced concrete with no voids, then it is called a one-way solid slab. 2.
One-way joist floor system: This type of
slab is also called a ribbed slab. It consists of a floor slab, usually
thick, supported
by reinforced concrete ribs (or joists). The ribs are usually tapered and are uniformly spaced at distances that do not exceed
. The ribs are supported on girders that rest on
columns. The spaces between the ribs may be formed using removable steel or fiberglass 186
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form fillers (pans), which may be used many times. In some ribbed slabs, the spaces between ribs may be filled with permanent fillers to provide a horizontal slab.
3.
Two-way floor systems: When the slab is supported on four sides and the ratio of
the long side to the short side is less than
(
), the slab will deflect in
double curvature in both directions. The floor load is carried in two directions to the four beams surrounding the slab. Other types of two-way floor systems are flat plate floors, flat slabs, and waffle slabs. This chapter deals only with one-way floor systems.
8.2 ANALYSIS OF CONTINUOUS BEAMS AND FRAMES.
In reinforced concrete structures, as much of the concrete as is practical is placed in one single operation. Reinforcing steel is not terminated at the ends of a member but is extended through the joints into adjacent members. At construction joints, special care is taken to bond the new concrete to the old by carefully cleaning the latter, by extending the reinforcement through the joint, and by other means. As a result, reinforced concrete structures usually represent monolithic, or continuous, units. A load applied at one location causes deformation and stress at all other locations. Even in precast concrete construction, which resembles steel construction in that individual members are brought to the job site and joined in the field, connections are often designed to provide for the transfer of moment as well as shear and axial load, producing at least partial continuity.
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In statically determinate structures, such as simple-span beams, the deflected shape and
the moments and shears depend only on the type and magnitude of the loads and the dimensions of the member. In contrast, inspection of the statically indeterminate structures shows that the deflection curve of any member depends not only on the loads but also on the joint rotations, whose magnitudes in turn depend on the distortion of adjacent, rigidly connected members. Continuous beams and frames can be analyzed using approximate methods or computer programs, which are available commercially. Other methods, such as the displacement and force methods of analysis based on the calculation of the stiffness and flexibility matrices, may also be adopted. Slope deflection and moment-distribution methods may also be used. These methods are explained in books dealing with the structural analysis of beams and frames. However, the ACI Code, Section 8.3, gives approximate coefficients for calculating the bending moments and shear forces in continuous beams and slabs. These coefficients were given in Section 8.3 of this Chapter. The moments obtained using the ACI coefficients will be somewhat larger than those arrived at by exact analysis. The limitations stated in the use of these coefficients must be met. In the structural analysis of continuous beams, the span length is taken from center to center of the supports, which are treated as knife-edge supports. In practice, the supports are always made wide enough to take the loads transmitted by the beam, usually the
moments acting at the face of supports. To calculate the design moment at the face of the
support, it is quite reasonable to deduct a moment equal to moment at the centerline of the support, where width.
from the factored
is the factored shear and is the column
According to ACI Code Section 8.9 – Span Length: 8.9.1 — Span length of members not built integrally with supports shall be considered as the clear span plus the depth of the member, but need not exceed distance between centers of supports. 8.9.2 — In analysis of frames or continuous construction for determination of moments, span length shall be taken as the distance center-to-center of supports. 8.9.3 — For beams built integrally with supports, design on the basis of moments at faces of support shall be permitted. 8.9.4 — It shall be permitted to analyze solid or ribbed slabs built integrally with supports, with clear spans not more than 3 m, as continuous slabs on knife edge supports with spans equal to the clear spans of the slab and width of beams otherwise neglected. The individual members of a structural frame must be designed for the worst combi nation of loads that can reasonably be expected to occur during its useful life. Internal moments, shears, and thrusts are brought about by the combined effect of dead and live loads, plus other loads, such as wind and earthquake. While dead loads are constant, live loads such as floor loads from human occupancy can be placed in various ways, some of which will result in larger effects than others.
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For structural analysis of continuous beam or rib to obtain the shear and moment diagrams, it shall be permitted according to ACI code, 8.11.2, to assume that the arrangement of live load is limited to combinations of: a. Factored dead load on all spans with full factored live load on two adjacent spans; and b. Factored dead load on all spans with full factored live load on alternate spans.
Span 1
Span2
Span 3
Load Case 1: ACI-8.11.2-a
LL DL
Load Case 2: ACI-8.11.2-a
LL DL
Load Case 3: ACI-8.11.2-b
LL
LL
DL
Load Case 4: ACI-8.11.2-b
LL DL 189
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From each case we get the Maximum moment: •
Maximum negative moment moment from load cases 1+2 (ACI-8.11.2-a)
•
Maximum positive moment moment from load cases 3+4 (ACI-8.11.2-b)
•
Envelope moment diagram from all possible load cases.
Moment Diagram from
Load Case 1
Moment Diagram from
Load Case 2
Moment Diagram from
Load Case 3
Moment Diagram from
Load Case 4
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Moment Diagrams of all Load cases
Envelope Moment Diagram from all Load cases
8.3 ANALYSIS AND DESIGN OF ONE-WAY SOLID SLABS. ACI CODE LIMITATIONS.
If the concrete slab is cast in one uniform thickness without any type of voids, it can be referred to as a solid slab. In a one-way stab nearly all the loading is transferred in the short direction, and the slab may be treated as a beam. A unit strip of slab, usually 1 m at right angles to the supporting girders, is considered a rectangular beam. The beam has a unit width with a depth equal to the thickness of the slab and a span length equal to the distance between the supports. A one-way slab thus consists of a series of rectangular beams placed side by side.
If the slab is one span only and rests freely on its supports, the maximum positive moment for a uniformly distributed load of
is
, where is is the span length
between the supports. If the same slab is built monolithically with the supporting beams or is continuous over several supports, the positive and negative moments are calculated by structural analysis or by moment coefficients as for continuous beams. The ACI Code, Section 8.3, permits the use of moment and shear coefficients in the case of two or more approximately equal spans. The maximum positive and negative moments and shears are computed from the following
expressions: 191
Reinforced Concrete II
where 193).
and
Dr. Nasr Abboushi
are moment and shear coefficients given in table below and figure (page
For all positive midspan moments, all shears and the negative moment at exterior
supports,
, is for the span under consideration. For the negative moment at interior
supports,
, shall be taken as
as defined in the figure above.
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The conditions under which the moment coefficients for continuous beams and slabs should be used can be summarized as follows:
1. Spans are approximately equal: Longer span 2. Loads are uniformly distributed.
(shorter span).
3. The ratio (live load/dead load) is less than or equal to 4. For slabs with spans less than or equal to
face of all supports is
.
, negative bending moment at
5. For an unrestrained discontinuous end, the coefficient is at end support and
at midspan.
6. Shear force at C is
and at the face of all other support is
7. The members are prismatic.
.
When these conditions are not satisfied, structural structural analysis is required. In structural analysis, the negative bending moments at the centers of the supports are calculated. The value that may be considered in the design is the negative moment at the face of the support, ACI 8.9.2, 8.9.3. The following limitations are specified by the ACI code:
Atypical imaginary strip 1m wide is assumed.
The minimum thickness of one-way slabs using grade 420 steel can be defined according to the ACI Code, 9.5.2.1, Table 9.5a, for solid slabs and for beams or ribbed one-way slabs .
ACI 9.5.2.1 – Minimum thickness stipulated in Table 9.5(a) shall apply for one-way construction not supporting or attached to partitions or other construction likely to be damaged by large deflections, unless computation of deflection indicates a lesser thickness can be used without adverse effects.
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Deflection is to be checked when the slab supports are attached to construction likely to be damaged by large deflections. Deflection limits are set by the ACI Code, Table 9.5b.
ACI 9.5.2.2 – Where deflections are to be computed, deflections that occur immediately on application of load shall be computed by usual methods or formulas for elastic deflections, considering effects of cracking and reinforcement on member stiffness.
It is preferable to choose slab depth to the nearest
Shear should be checked, although it does not usually control.
Concrete cover in slabs shall not be less than weather or ground. In this case,
.
at surfaces not exposed to
.
In structural slabs of uniform thickness, the minimum amount of reinforcement in the direction of the span shall not be less than that required for shrinkage and temperature reinforcement (ACI Code, Section 7.12).
slab thickness nor more than
The principal reinforcement shall be spaced not farther apart than three times the (ACI Code, Section 7.6.5).
Straight-bar systems may be used in both tops and bottoms of continuous slabs. An alternative bar system of straight and bent (trussed) bars placed alternately may also be used.
In addition to main reinforcement, steel bars at right angles to the main must be propro vided. This additional steel is called secondary, distribution, shrinkage, or temperature reinforcement. 8.4 MINIMUM REINFORCEMENT RATIO.
For structural slabs and of uniform thickness,
in the direction of the span shall be the
same as that required by 7.12.2.1 for temperature and shrinkage reinforcement (see section 8.5).
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Maximum spacing of this reinforcement shall not exceed three times the thickness, nor .
To limit the widths of flexural cracks in beams and slabs, ACI Code Section 10.6.4 defines upper limit on the center-to-center spacing between bars in the layer of reinforcement closest to the tension face of a member. The spacing limit is:
but
where
is the least distance from
surface of reinforcement to the tension face. It shall be permitted to take
.
as
8.5 TEMPERATURE AND SHRINKAGE REINFORCEMENT.
Concrete shrinks as the cement paste hardens, and a certain amount of shrinkage is usually anticipated. If a slab is left to move freely on its supports, it can contract to accommodate the shrinkage. However, slabs and other members are joined rigidly to other parts of the structure, causing a certain degree of restraint at the ends. This results in tension stresses known as shrinkage stresses. A decrease in temperature and shrinkage stresses is likely to cause hairline cracks. Reinforcement is placed in the slab to counteract contraction and distribute the cracks uniformly. As the concrete shrinks, the steel bars are subjected to compression. Reinforcement for shrinkage and temperature stresses normal to the principal reinforcement should be provided in a structural slab in which the principal reinforcement extends in one direction only. The ACI Code, Section 7.12.2, specifies that: area of shrinkage and temperature
×
reinforcement shall provide at least the following ratios of reinforcement area to gross concrete area, but not less than
:
For slabs in which grade 280 (
are used,
) or 350 (
.
For slabs in which grade 420 (
welded wire fabric are used,
) deformed bars or welded bars or
.
For Slabs where reinforcement with yield stress exceeding
yield strain of
) deformed bars
percent is used,
measured at a
Shrinkage and temperature reinforcement shall be spaced not farther apart than five times the slab thickness, nor farther apart than
.
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For temperature and shrinkage reinforcement, the whole concrete depth shrinkage shall be used to calculate the steel area.
exposed to
8.6 REINFORCEMENT DETAILS.
In continuous one-way slabs, the steel area of the main reinforcement is calculated for all critical sections, at midspans, and at supports. The choice of bar diameter and detailing depends mainly on the steel areas, spacing requirements, and development length. Two bar systems may be adopted.
In the straight-bar system: straight bars are used for top and bottom reinforcement
in all spans. The time and cost to produce straight bars is less than that required to produce bent bars; thus, the straight-bar system is widely used in construction.
In the bent-bar, or trussed, system : straight and bent bars are placed alternately in
the floor slab. The location of bent points should be checked for flexural, shear, and development length requirements. For normal loading in buildings, the bar details at the end and interior spans of one-way solid slabs may be adopted as shown in figures.
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Example:
Design a simply supported one-way solid slab, span of (
). Dead Load – own weight only.
, subjected to service live load and
.
Solution: Minimum thickness (deflection requirements). For simply supported one-way solid
slab:
Assume bar diameter
for main reinforcement.
Loads calculation:
s
r e v o C
Use
Provide
then
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× – – × – –
Take Step
.
is the smallest of:
1. 2.
Take
.
Step
is the smallest of:
1. 2.
Example:
The cross-section of a continuous one-way solid slab in a building is shown below. The slabs are supported by beams that span
between simple supports. The dead load on the
slabs is that due to self-weight plus
; the live load is
continuous slab and draw a detailed section. Given:
201
and
. Design the .
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Dr. Nasr Abboushi
Solution: Minimum thickness (deflection requirements).
Take slab thickness
Assume bar diameter
for main reinforcement.
Loads calculation:
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Check whether thickness is adequate for shear:
× × √ × × - for shear.
The thickness of the slab is adequate enough.
Even, if
for solid slabs, the thickness of the slab will be enough .
Factored moments at sections A, B, C, D, E:
For the negative moment at interior supports,
, shall be taken as
.
Here
Location
A B C
D E
Slab Design for the positive moments:
Midspan section B:
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× – – – – Use
then
Take Step
.
is the smallest of:
1. 2.
Midspan section E:
Use
then
Take
.
Slab Design for the negative moments:
Note that the second support has two values of moments by analysis, at section C and
section D. In construction, the provided reinforcement will be the same bar diameters on opposite sides of the support, so the design may be done for the maximum moment of the two moments at C and D (Only one design for Support section C). 204
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– – – – Support section C:
Assume bar diameter
Use
for main reinforcement.
then
Take
.
Support section D (interior D supports):
Assume bar diameter
Use
for main reinforcement.
then
Take
.
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– – × –– Support section A:
Assume bar diameter
for main reinforcement.
Use
Provide
then
Take
.
Temperature and shrinkage reinforcement.
Take
.
Step
is the smallest of:
1. 2.
Required
Location
A
3.3
B
3.3
C and first interior D
3.3
Interior D
3.3
E
3.3
Provided
Temperature and shrinkage reinforcement
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Reinforcement
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Dr. Nasr Abboushi
Example:
Design the one-way solid slab, which plan is shown below. The dead load on the slabs is that
due to self-weight plus weight of: • • • • •
Tiles,
.
Mortar, Sand,
.
.
Plaster,
.
Partitions,
The live load is
.
.
Given: and
.
Solution:
Minimum
thickness
(deflection requirements). From the maximum span length
for
one-end
continuous, we get
Take slab thickness
Assume bar diameter
for main reinforcement.
Loads calculation:
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Tiles mortar Sand Reinforced Concrete solid slab Plaster Partitions
Quality Density
Material
Total Dead Load
Dead Load for 1 m strip of slab Live Load for 1 m strip of slab
208
× × × × ×
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Dr. Nasr Abboushi
According to ACI 8.9.2 — In analysis of frames or continuous construction for determination of moments, span length shall be taken as the distance center-to-center of supports. According to ACI 8.9.3 — For beams built integrally with supports, design on the basis of moments at faces of support shall be permitted. Check whether thickness is adequate for shear:
√ - for shear.
The thickness of the slab is adequate enough.
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Slab Design for the negative moment:
×× × – – Assume bar diameter
Use
for main reinforcement.
then
Take Step
is the smallest of:
1. 2.
Check for strain (tension-controlled section -
210
):
.
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Dr. Nasr Abboushi
Slab Design for the positive moments:
– – – – Assume bar diameter
Use
for main reinforcement.
then
Take
Assume bar diameter
Use
for main reinforcement.
then
Take
For both positive moment designs
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Temperature and shrinkage reinforcement.
× ––
Take
.
Step
is the smallest of:
1. 2.
8.7
ONE-WAY JOIST FLOORS AND ONE-WAY RIBBED SLABS.
A one-way joist floor system consists of hollow slabs with a total depth greater than that of solid slabs. The system is most economical for buildings where superimposed loads are small and spans are relatively large, such as schools, hospitals, and hotels. The concrete in the tension zone is ineffective; therefore, this area is left open between ribs or filled with lightweight material to reduce the self-weight of the slab. 212
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The design procedure and requirements of ribbed slabs follow the same steps as those for rectangular and T-sections explained in Chapter 4. The following points apply to design of one-way ribbed slabs:
1. Ribs are usually tapered and uniformly spaced at about usually formed by using pans (molds)
wide and
. Voids are deep,
depending on the design requirement. The standard increment in depth is 50 mm.
2. The ribs shall not be less than
wide and must have a depth of not more than
times the width. Clear spacing between ribs shall not exceed
Section 8.13).
213
(ACI Code,
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Dr. Nasr Abboushi
3. Shear strength,
, provided by concrete for the ribs may be taken
greater than
that for beams. This is mainly due to the interaction between the slab and the closely spaced ribs (ACI Code, Section 8.13.8). 4. The thickness of the slab on top of the ribs is usually
and contains
minimum reinforcement (shrinkage reinforcement). This thickness over permanent fillers shall not be less than
of the clear span between ribs or
(ACI Code,
Section 8.13.5.2). When removable forms or fillers are used slab thickness shall be not less than
of the clear distance between ribs, nor less than
Code, Section 8.13.6.1)
. (ACI
5. The ACI coefficients for calculating moments in continuous slabs can be used for continuous ribbed slab design. If the live load on the ribbed slab is less than
and the span of ribs exceeds
,
a secondary transverse rib (distribution rib) should be provided at midspan (its direction is perpendicular to the direction of main ribs) and reinforced with the same amount of steel as the main ribs. Its top reinforcement shall not be less than half of the main reinforcement in
the tension zone. These transverse ribs act as floor stiffeners. If the live load exceeds and the span of ribs varies between and
as indicated before. If the span exceeds
, one traverse rib must be provided,
, at least two transverse ribs at one-third span
must be provided with reinforcement, as explained before.
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Reinforcement for the joists usually consists of two bar in the positive bending region, with one bar discontinued where no longer needed or bent up to provide a part of the negative steel requirement over supporting girder. According to ACI Code section 7.13.2 at least one bottom bar must be continuous over the support, or at non continuous supports, terminated in a standard hook, as a measure to improve structural integrity in the event of major structural damage. The minimum thickness of beams or ribbed one way slabs depending on the support conditions can be determined according to ACI Code 9.5.2. (see table 9.5(a), page 194). 8.8 DESIGN OF ONE-WAY RIBBED SLAB.
For the ribbed slab plan with section as shown below: •
Determine the total slab thickness.
•
Design the topping slab.
•
Design the rib for flexure and shear, the envelope moment and shear diagrams are shown.
•
Design the beams B1, B2 for flexure and shear, the envelope moment and shear diagrams are shown.
•
Take the material's density from the table below.
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Quality Density
Material Tiles mortar Sand
Reinforced Concrete Hollow Block Plaster Partitions
Compressive strength of concrete Yield strength of steel, Live Load,
.
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Solution:
800 mm
Rib 2 800 mm
4 m a e B
3
2
a e B
a e B
a
Rib 1
800 mm
800 mm
800 mm
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800 mm
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Minimum thickness (deflection requirements).
There are two groups of ribs and beams (Rib 1; Rib 2; Beam 1, 2, 3, 4; Beam 5, 6). The thickness of the one-way ribbed slab without drop beams can be obtained according to
ACI code, table 9.5 (a).
The maximum span length for one-end continuous (for ribs):
then then
The maximum span length for both-ends continuous (for ribs):
then then
The maximum span length for one-end continuous (for Beams):
then then
The maximum span length for both-ends continuous (for Beams):
then then
The minimum ribbed slab thickness will be Take slab thickness
Topping Design.
Topping in One way ribbed slab can be
considered as a strip of 1 meter width and
span of hollow block length with both end fixed in the ribs. Dead Load calculations:
×
Dead Load from:
Tiles Mortar Coarse Sand Topping Interior Partitions
×× × × × ×
Live Load calculations: Total Factored Load:
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√ √ × Strength condition, where
for plain concrete.
(ACI 22.5.1, Equation 22-2)
where
for rectangular rectangular section of the slab:
NO Reinforcement is required by analysis. According to ACI 10.5.4., provide as shrinkage and temperature reinforcement.
for slabs
× – – According to ACI 7.12.2.1,
.
Try bars 8 with
Take Step
in both directions.
is the smallest of:
1. 2.
Take
in both directions.
From practical concederation, the secondary reinforcement parallel to the ribs shall be placed in the slab and spaced at distances not more than half of the spacings between ribs (usually two bars upon each
width block).
Load Calculations for Rib 1:
From the Geometry of T-section:
220
= t m m 0 5 2 = h
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Dead Load calculations: Dead Load from: Tiles Mortar Live Load calculations:
Coarse Sand
×
Topping RC Rib Hollow Block Plaster
×× ×× ×× ×× ×× ×× ×× ×× ×
Interior Partitions
Dead Load / rib:
Live Load /rib:
The Effective Flange width (
) According to ACI 8.12.2 (see page 220):
is the smallest of:
Take
.
Structural Analysis of Rib 1. The envelope shear and moment diagrams (for all load
combinations). Using the structural analysis and design programs, we obtain the Envelope Moment diagram for Rib1.
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Dr. Nasr Abboushi
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Dr. Nasr Abboushi
̅ × ̅ × √ Design of Rib 1 for positive moments.
Assume bar diameter
for main positive reinforcement.
The maximum positive moment in all spans of Rib 1
Check if
The section will be designed as rectangular section with
.
Check for
Use
with
Check for strain:
Usually, no reinforcement less than moments equal or less than
can be used. So, for all spans with positive
, use
223
for each rib span.
Reinforced Concrete II
Dr. Nasr Abboushi
Design of Rib 1 for ne gative moments.
Assume bar diameter
for main positive reinforcement.
According to ACI 8.9.3 — for beams built integrally with supports, design on the basis of moments at faces of support shall be permitted. The maximum negative moment at the face of support
× Check for
Use
with
Check for strain:
Usually, no reinforcement less than
can be used. So, for all supports with negative
moments equal or less than
, use
for each rib support.
Design of Rib 1 for shear.
The maximum shear force at the distance from the face of support Shear strength,
, provided by concrete for the ribs may be taken
greater than that for
beams. This is mainly due to the interaction between the slab and the closely spaced ribs (ACI Code, Section 8.13.8). 224
Reinforced Concrete II
Dr. Nasr Abboushi
× × √
Minimum shear reinforcement is required except for concrete joist construction. So, No shear reinforcement is provided.
Load calculations for Beam 4:
The distributed Dead and Live loads acting upon the Beam 4 can be defined from the support support reactions of the rib 1 and rib 2. 2.
Beam 4
Dead Load calculations:
The maximum support reaction (factored) from Dead Loads for rib1 upon beam 4 is
. The distributed Dead Load from the Rib 1 on Beam 4:
225
Reinforced Concrete II
Dr. Nasr Abboushi
Assume the width of the beam
, then the own weight of the beam and the weight
of the floor layers within the beam width can be calculated:
×× ×× ×× ×× ×× ×× ×
Dead Load from:
Tiles Mortar Coarse Sand RC Beam Plaster Interior Partitions
The total factored Dead Load: Live Load calculations:
The maximum support reaction (factored) from Live Loads for rib1 upon beam 4 is The distributed Live Load from the Rib 1 on Beam 4:
×
The Live Load within the beam width (
The total factored Live Load:
.
) can be calculated:
Important NOTE:
The dead and live loads acting within the beam width have been calculated twice actually. That because the support reactions of ribs from the dead and live loads acting over the ribs were calculated for full span of the ribs, measured center to center, which include the whole beam width. More accurately calculations to determine the loads on the beam could be done by taking the loads that transfer from the rib to the beam which could be calculated as the sum of shear values of the ribs at the face of support (beam) from each side and then adding the dead and live loads acting directly on the beam within the beam width. From the next envelope shear diagrams, the shear values at the face of beam 4 are:
From dead load:
from the left, left, and
from the right) which means
that the total dead load that transfers to the beam 4 is
The uniformly distributed dead load over the beam from the ribs only is
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Reinforced Concrete II
From live load:
Dr. Nasr Abboushi
from the left, and
from the right) which means
that the total live load that transfers to the beam 4 is
The uniformly distributed live load over the beam from the ribs on ly is
Shear diagram from Dead load only only .
Beam 4
Envelope shear diagram from Live load only only .
Beam 4
The total factored Dead Load: The total factored Live Load:
227
Reinforced Concrete II
Dr. Nasr Abboushi
I’ v h h f h h f ll. I’ fl
to calculate the loads more accurate specially when the beam section is wide and the spans are large. We can use the first method for simplicity, especially in this stage of calculations, when the dimensions of the beam are not known yet.
Structural Analysis of Beam 4. The envelope shear and moment diagrams (for all load
combinations). The Beam 4 is loaded from the ribs 1 (first two spans) and ribs 2 (last span). The load transferred from ribs 1 to Beam 4 is calculated before. The load transferred from rib 2 to Beam 4 will be obtained by analyzing the rib 2 as continuous beam as follows:
Beam 4
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Reinforced Concrete II
Dr. Nasr Abboushi
Dead Load calculations:
The maximum support reaction (factored) from Dead Loads for rib2 upon beam 4 is
. The distributed Dead Load from the Rib 1 on Beam 4:
The total factored Dead Load: Live Load calculations:
The maximum support reaction (factored) from Live Loads for rib 2 upon beam 4 is The distributed Live Load from the Rib 2 on Beam 4:
.
The total factored Live Load:
Using the structural analysis and design programs, we o btain the Envelope Moment diagram for Beam 4.
229
Reinforced Concrete II
Dr. Nasr Abboushi
Design of Beam 4 for flexure.
Assume bar diameter
for main positive reinforcement. reinforcement.
The width of the Beam 4 can be defined from the maximum factored moment. The maximum factored moment in Beam 4
Note that according to ACI 8.9.3 — for beams built integrally with supports, design on the
basis of moments at faces of support shall be permitted. Here the design will be done for the moments at the center of supports. Take
Assume
f fl -ll
. 230
Reinforced Concrete II
Dr. Nasr Abboushi
Take
(
)
Usually in construction the maximum width of the beams is
. Here, take
and no need to recalculate the loads acting on the beam.
Note that the factored moments of other supports and spans may be satisfied by the section
width of
as a singly reinforced beam sections, but the support section with
may be designed as doubly reinforced section.
Check whether the section will be act as singly or
doubly reinforced section: Maximum nominal moment strength from strain
× condition
Design the section as singly reinforced concrete section.
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Dr. Nasr Abboushi
√ ××× Take
in one layer with
Check for strain:
Check for bar placement:
Design of Beam 4 for shear.
√ √ Critical section at distance
from the face of support.
Check for section dimensions:
Find the maximum stirrups spacing:
232
Reinforced Concrete II
Dr. Nasr Abboushi
√ √ ( ) Check for
:
Or
Compute the stirrups spacing required to resist the shear forces: Use stirrups 2U-shape (4 legs stirrups) with
Take 2U-shape (4 legs stirrups)
100 cm
233
m c 5 2
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Dr. Nasr Abboushi
Design the other beam sections for flexure (for positive and negative moments and caculate the area of steel for each section).
Note that for shear design, it is obvious that, the stirrups cannot be less than two U-shape
stirrups and the step
for all sections where stirrups are reqired.
So, for all sections the design for shear will be as the previous section ( 2 U-shape
).
An alternative design for the Beam 4 can be done on the b asis of drop beam section, not as a hidden beam as in the previous design.
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Reinforced Concrete II
CHAPTER 9
Dr. Nasr Abboushi
TWO-WAY SLABS
9.1 INTRODUCTION
When the slab is supported on all four sides and the length, , is less than twice the width, , the slab will deflect in two directions, and the loads on the slab are transferred to all four
supports. This slab is referred to as a two-way slab. The bending moments and deflections in such slabs are less than those in one-way slabs; thus, the same slab can carry more load when supported on four sides. The load in this case is carried in two directions, and the bending moment in each direction is much less than the bending moment in the slab if the load were carried in one direction only.
9.2 TYPES OF TWO-WAY SLABS
Structural two-way concrete slabs may be classified as follows: 1. Two-Way Slabs on Beams: This case occurs occurs when the two-way slab slab is supported by beams on all four sides. The loads from the slab are transferred to all four supporting beams, which, in turn, transfer the loads to the columns. 2. Flat Slabs: A flat slab is a two-way slab reinforced in two directions that usually does not have beams or girders, and the loads are transferred directly to the supporting columns. The column lends to punch through the slab, which can be treated by three methods: 234
Reinforced Concrete II
Dr. Nasr Abboushi
a. Using a drop panel and a column capital. b. Using a drop panel without a column capital. The concrete panel around the column capital should be thick enough to withstand the diagonal tensile stresses arising from the punching shear. c. Using a column capital without drop panel, which is not common. 3. Flat-Plate Floors: A flat-plate floor is a two-way slab system consisting of a uniform slab that rests directly on columns and does not have beams or column capitals (Fig. a). In this case the column tends to punch through the slab, producing diagonal tensile stresses. Therefore, a general increase in the slab thickness is required or special reinforcement is used.
235
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Dr. Nasr Abboushi
4. Two-Way Ribbed Slabs and the Waffle Slab System : This type of slab consists of a floor slab with a length-to-width ratio less than 2. The thickness of the slab is usually
5 to 10 cm and is supported by ribs (or joists) in two directions. The ribs are arranged in each direction at spacings of about
rectangular shapes. The ribs can also be arranged at
, producing square or
or
from the centerline
of slabs, producing architectural shapes at the soffit of the slab. In two-way ribbed slabs, different systems can be adopted: a.
A two-way rib system with voids between the ribs, obtained by using special removable and usable forms (pans) that are normally square in shape. The ribs are supported on four sides by girders that rest on columns. This type is called a twoway ribbed (joist) slab system.
b.
A two-way rib system with permanent fillers between ribs that produce horizontal slab soffits. The fillers may be of hollow, lightweight or normal-weight concrete or any other lightweight material. The ribs are supported by girders on four sides, which in turn are supported by columns. This type is also called a two-way ribbed (joist) slab system or a hollow-block two-way ribbed system.
c.
A two-way rib system with voids between the ribs with the ribs continuing in both directions without supporting beams and resting directly on columns through solid panels above the columns. This type is called a waffle slab system.
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Dr. Nasr Abboushi
9.3 ECONOMICAL CHOICE OF CONCRETE FLOOR SYSTEMS
Various types of floor systems can be used for general buildings, such as residential, office, and institutional buildings. The choice of an adequate and economic floor system depends on the type of building, architectural layout, aesthetic features, and the span length
between columns. In general, the superimposed live load on buildings varies between 4 and . A general guide for the economical use of floor systems can be summarized as
follows:
1. Flat Plates: Flat plates plates are most suitable suitable for spans of between and
and live loads
. The advantages of adopting flat plates include low-
cost formwork, exposed flat ceilings, and fast construction. Flat plates have low shear capacity and relatively low stiffness, which may cause noticeable deflection. Flat plates are widely used in buildings either as reinforced or prestressed concrete slabs. 237
Reinforced Concrete II
Dr. Nasr Abboushi
2. Flat Slabs: Flat slabs are most suitable for spans of
and for live loads of
. They need more formwork than flat plates, especially for column
capitals. In most cases, only drop panels without column capitals are used.
3. Waffle Slabs: Waffle slabs are suitable for spans of
and live loads of
. They carry heavier loads than flat plates and have attractive
exposed ceilings. Formwork, including the use of pans, is quite expensive.
4. Slabs on Beams: Slabs on beams are suitable for spans beiween live loads of
and
and
. The beams increase the stiffness of the slabs,
producing relatively low deflection. Additional formwork for the beams is
needed.
5. One-Way Slabs on Beams: One-way slabs on beams are most suitable suitable for spans of
and a live load of
. They can be used for larger spans
with relatively higher cost and higher slab deflection. Additional formwork for the beams is needed.
6. One-Way Joist Floor System: A one-way joist floor system is most suitable for spans of
and live loads of
. Because of the deep ribs, the
concrete and steel quantities are relatively low, but expensive formwork is expected. The exposed ceiling of the slabs may look attractive.
9.4 MINIMUM THICKNESS OF TWO-WAY SLABS.
The ACI Code, Section 9.5.3, specifies a minimum slab thickness in two-way slabs to control
deflection. The magnitude of a slab's deflection depends on many variables, including the flexural stiffness of the slab, which in turn is a function of the slab thickness, . By increasing the slab thickness, the flexural stiffness of the slab is increased, and consequently the slab deflection is reduced. Because the calculation of deflections in two-way slabs is complicated and to avoid excessive deflections, the ACI Code limits the thickness of these slabs by adopting the following three empirical limitations, which are based on experimental
( )
research. If these limitations are not met, it will be necessary to compute deflections. 1. For
but not less than 2. For
but not less than 3. For
.
.
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Reinforced Concrete II
Dr. Nasr Abboushi
where
clear span in the long direction direction measured face to face of columns (or face to face of beams for slabs with beams).
the ratio of the long to the short clear spans. the average value of
for all beams on the sides of a panel.
the ratio of flexural stiffness of a beam section
to the flexural stiffness of the slab
, bounded laterally by the centerlines of the panels on each side of the beam.
where
, and
are the moduli of elasticity of concrete in the beam and the slab,
respectively, and
the gross moment of inertia of the beam section about the centroidal axis (the beam section includes a slab length on each side of the beam equal to the projection of the beam above or below the slab, whichever is greater, but not more than four times the
slab thickness) the moment of inertia of the gross section of the slab.
However, the thickness of any slab shall not be less than the following: 1. For slabs with 2. For slabs with
then thickness
then thickness
If no beams are used, as in the case of flat plates, then equations for calculating slab thickness, the panel shape, the steel
, and the flexural stiffness
of beams. When very stiff
beams are used, Eq. ( ) give
a
small
thickness, and Eq. (
slab ) may
control. For flat plates and flat slabs, when no interior beams minimum
are
used,
slab
. The ACI Code
, take into account the effect of the span length,
reinforcement yield stress,
may
and
the
thickness
may be determined directly from Table 9.5(c) of the ACI Code, which is shown here.
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Reinforced Concrete II
Dr. Nasr Abboushi
Other ACI Code limitations are summarized as follows: 1. For panels with discontinuous edges, end beams with a minimum
equal to
be used; otherwise, the minimum slab thickness calculated by Eqs. ( ) and ( increased by at least
(ACI Code, Section 9.5.3).
must
) must be
2. When drop panels are used without beams, the minimum slab thickness may be reduced by
. The drop panels should extend in each direction from the centerline of
support a distance not less than one-sixth of the span length in that direction between center to center of supports and also project below the slab at least
is included in Table 9.5(c).
3. Regardless of the values obtained by Eqs. ( ) and ( shall not be less than the following:
. This reduction
), the thickness of two-way slabs
(1) for slabs without beams or drop panels,
;
(2) for slabs without beams but with drop panels,
;
(3) for slabs with beams on all four sides with
,
for
,
, and
(ACI Code, Section 9.5.3.).
The thickness of a slab also may be governed by shear. This is particularly likely if large moments are transferred to edge columns and for interior columns between two spans that are greatly different in length. The selection of slab thicknesses to satisfy shear requirements
will be discussed later. Briefly, it is suggested that the trial slab thickness be chosen such that at edge columns and
240
at interior columns.
Reinforced Concrete II
Dr. Nasr Abboushi
9.5 SLAB REINFORCEMENT REQUIREMENTS. Placement Sequence.
In a flat plate or flat slab, the moments are larger in the slab strips spanning the long direction of the panels. As a result, the reinforcement for the long span generally is placed
closer to the top and bottom of the slab than is the short-span reinforcement. This gives the larger effective depth for the larger moment. For slabs supported on beams having greater than about
, the opposite is true, and the reinforcing pattern should be reversed.
If a particular placing sequence has been assumed in the reinforcement design, it should be shown or noted on the drawings. It also is important to maintain the same arrangements of layers throughout the entire floor, to avoid confusion in the field. Thus, if the east –west reinforcement is nearer the top and bottom surfaces in one area, this arrangement should be maintained over the entire slab, if at all possible. Concrete Cover.
ACI Code Section 7.7.1 specifies the minimum clear cover to the surface of the reinforcement in slabs as
for
and smaller bars, bars, provided that the slab is not not
exposed to earth or to weather.
Spacing Requirements, Minimum Reinforcement, and Minimum Bar Size.
ACI Code Section 13.3.1 requires that the minimum area of reinforcement provided for
flexure should not be less than (see page p age 195): For slabs in which grade 280 (
are used,
) or 350 (
) deformed bars
.
For slabs in which grade 420 (
welded wire fabric are used,
) deformed bars or welded bars or
.
The maximum spacing of reinforcement at critical design sections for positive and negative moments in both the middle and column strips shall not exceed two times the slab thickness (ACI Code Section 13.3.2), and the bar spacing shall not exceed
at any location (ACI
Code Section 7.12.2.2).
Although there is no code limit on bar size, the top steel bars abd steps in slab should be enough to give adequate rigidity to prevent displacement of the bars under ordinary foot traffic before the concrete is placed. Bar Cutoffs and Anchorages
For slabs without beams, ACI Code Section 13.3.8.1 allows the bars to be cut off as shown in the figure below (ACI Code Fig. 13.3.8). Where adjacent spans have unequal lengths, the extension of the negative-moment bars past the face of the support is based on the length of the longer span. ACI Code Section 13.3.3 requires that the Positive moment reinforcement perpendicular to a
discontinuous edge shall extend to the edge of slab and have embedment, straight or hooked, at least
in spandrel beams, columns, or walls. 241
Reinforced Concrete II
Dr. Nasr Abboushi
ACI Code Section 13.3.4 requires that all negative-moment steel perpendicular to an edge be
bent, hooked, or otherwise anchored in spandrel beams, columns, and walls along the edge to develop
in tension. If there is no edge beam, this steel still should be hooked to act as
torsional reinforcement and should extend to the minimum cover thickness from the edge of the slab.
ACI 318 - Fig. 13.3.8 — Minimum Minimum extensions for reinforcement in slabs without beams. (See 12.11.1 for reinforcement extension into supports).
ACI Code Section 13.3.6 requires that at exterior corners of slabs supported by edge walls or where one or more edge beams have a value of
greater than
, top and bottom slab
reinforcement shall be provided at exterior corners in accordance with 13.3.6.1 through 13.3.6.4. 13.3.6.1 — Corner reinforcement in both top and bottom of slab shall be sufficient to
resist a moment per unit of width equal to the maximum positive moment per unit width in the slab panel. 13.3.6.2 — The moment shall be assumed to be about an axis perpendicular to the
diagonal from the corner in the top of the slab and about an axis parallel to the diagonal from the corner in the bottom of the slab. 13.3.6.3 — Corner reinforcement shall be provided for a distance in each direction
from the corner equal to one-fifth the longer span. 242
Reinforced Concrete II
Dr. Nasr Abboushi
13.3.6.4 — Corner reinforcement shall be placed parallel to the diagonal in the top of
the slab and perpendicular to the diagonal in the bottom of the slab. Alternatively, reinforcement shall be placed in two layers parallel to the sides of the slab in both the top and bottom of the slab.
9.6 SHEAR STRENGTH OF TWO-WAY SLABS.
In a two-way floor system, the slab must have adequate thickness to resist both bending moments and shear forces at the critical sections. To investigate the shear capacity of twoway slabs, the following cases should be considered. 243
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Dr. Nasr Abboushi
9.6.1 Two-Way Slabs Supported on Beams
In two-way slabs supported on beams, the critical sections are at a distance of
the
supporting
beams,
and
the
shear
capacity
of
each
from the face section
is
. When the supporting beams are stiff and are capable of transmitting
floor loads to the columns, they are assumed to carry loads acting on floor areas bounded by
lines drawn from the corners, as shown in the figure below. The loads on the trapezoidal
areas will be carried by the long beams
areas will be carried by the short beams
highest between
and
and
, whereas the loads on the triangular . The shear per unit width of slab is
in both directions, and
factored load per unit area.
and
, where
is the uniform
If no shear reinforcement is provided, the shearing force at a distance from the face of the beam,
where
, must be equal to
9.6.2 Two-Way Slabs Without Beams
In flat plates and flat slabs, beams are not provided, and the slabs are directly supported by
columns. In such slabs, two types of shear stresses must be investigated; the first is one-way shear, or beam shear. The critical sections are taken at a distance 244
from the face of the
Reinforced Concrete II
Dr. Nasr Abboushi
column, and the slab is considered as a wide beam spanning between supports, as in the case of one-way beams. The shear capacity of the concrete section is
.
The second type of shear to be studied is twoway, or punching, shear, as in the design of footings. Shear failure occurs along a truncated
cone or pyramid around the column. The critical section is located at a distance
from the
face of the column, column capital, or drop
panel. The ACI Code, Section 11.11.2 allows a shear strength,
, in slabs and footings without
shear reinforcement for two-way shear action, the smallest of
where
Ratio of long side to short side of o f the rectangular column.
For shapes other than rectangular,
is taken to be the ratio of the longest dimension of the
effective loaded area in the long direction to the largest width in the short direction
(perpendicular to the long direction).
perimeter of the critical section taken at
effective depth of slab.
for normal-weight concrete.
is assumed to be:
for interior columns,
for edge columns, and
for corner columns. 245
from the loaded area.
Reinforced Concrete II
Dr. Nasr Abboushi
Edge of the slab
When openings are located at less than
times the slab
thickness from a column, ACI Code Section 11.11.6 requires that the critical perimeter be reduced, as shown below. ACI
Commentary
suggested
that the side faces of the critical perimeter would extend to the edge of the slab if the distance from the face of the column to the edge of the slab
does not exceed the larger of: (i) four slab thicknesses, (ii) twice length,
the
, or
development
, of the flexural
reinforcement perpendicular to the edge, shown by the distances labeled A and B in the Figures (b) and (c). 246
b a l s e h t f o e g d E
Edge of the slab
Reinforced Concrete II
Dr. Nasr Abboushi
9.6.3 Tributary Areas for Shear in Two-Way Slabs.
For uniformly loaded two-way slabs, the tributary areas used to calculate are bounded by
lines of zero shear. For interior panels, these lines can be assumed to pass through the
center of the panel. For edge panels, lines of zero shear are approximately at from the center of the exterior column, where
to to
is the span measured from center-to-
center of the columns. However, to be conservative in design, ACI Code Section 8.3.3
requires that the exterior supports must resist a shear force due to loads acting on half of the span
. Also, to account for the larger tributary area for the first interior support, ACI
Code Section 8.3.3 requires that the shear force from loads acting on half of the span must be increased by
. This essentially results in a tributary length of
.
Critical sections and tributary areas for shear in a flat plate. 9.6.4 Shear Reinforcement in Two-Way Slabs Without Beams.
In flat-slab and flat-plate floor systems, the thickness of the slab selected may not be adequate to resist the applied shear stresses. In this case, either the slab thickness must be increased or shear reinforcement must be provided. The ACI Code allows the use of shear reinforcement by shearheads and anchored bars or wires (see next figures). The design for shear (punching) reinforcement in flat plates will be discussed later in details (see section 9.11). 247
Reinforced Concrete II
Dr. Nasr Abboushi
248
Reinforced Concrete II
Dr. Nasr Abboushi
9.7 ANALYSIS AND DESIGN OF TWO-WAY SLABS.
An exact analysis of forces and displacements in a two-way slab is complex, due to its highly indeterminate nature; this is true even when the effects of creep and nonlinear behavior of the concrete are neglected. Numerical methods such as finite elements can be used, but 249
Reinforced Concrete II
Dr. Nasr Abboushi
simplified methods such as those presented by the ACI Code are more suitable for practical design. The ACI Code, Chapter 13, assumes that the slabs behave as wide, shallow beams that form, with the columns above and below them, a rigid frame. The validity of this assumption of dividing the structure into equivalent frames has been verified by analytical and experimental research. It is also established that factored load capacity of two-way slabs with restrained boundaries is about twice that calculated by theoretical analysis, because a great deal of moment redistribution occurs in the slab before failure. At high loads, large deformations and deflections are expected; thus, a minimum slab thickness is required to maintain adequate deflection and cracking conditions under service loads. The ACI Code specifies two methods for the design of two-way slabs: 1. The direct design method. DDM (ACI Code, Section 13.6), is an approximate procedure for the analysis and design of two-way slabs. It is limited to slab systems subjected to uniformly distributed loads and supported on equally or nearly equally spaced columns. The method uses a set of coefficients to determine the design moments at critical sections. Two-way slab systems that do not meet the limitations of the ACI Code, Section 13.6.1, must be analyzed by more accurate procedures. 2. The equivalent frame method, EFM (ACI Code, Section 13.7), is one in which a threedimensional building is divided into a series of two-dimensional equivalent frames by cutting the building along lines midway between columns. The resulting frames are considered separately in the longitudinal and transverse directions of the building and treated floor by floor, as will be shown later. The systems that do not meet the requirements permitting analysis by the "direct design method" of the present code, has led many engineers to continue to use the design method of the 1963 ACI Code (The coefficient method) for the special case of two-way slabs supported on four sides of each slab panel by relatively deep, stiff, edge beams . It has been
used extensively here since 1963 for slabs supported at the edges by walls, steel beams, or monolithic concrete beams having a total depth not less than about 3 times the slab thickness. While it was not a part of the 1977 or later ACI Codes, its continued use is
permissible under the current code provision (ACI Code 13.5.1) that a slab system may be designed by any procedure satisfying conditions of equilibrium and geometric compatibility, if it is shown that the design strength at every section is at least equal to the required strength, and that serviceability requirements are met. 9.8 SLAB ANALYSIS BY THE COEFFICIENT METHOD.
The coefficient method makes use of tables of moment coefficients for a variety of conditions. These coefficients are based on elastic analysis but also account for inelastic redistribution. In consequence, the design moment in either direction is smaller by an appropriate amount than the elastic maximum moment in that direction. The moments in the middle strips in the two directions are computed from 250
Reinforced Concrete II
Dr. Nasr Abboushi
and
where
tabulated moment coefficients. uniform load,
length of clear span in short and long directions respectively.
The method provides that each panel be divided in both directions into a middle strip whose width is one-half that of the panel and two edge or column strips of one-quarter of the panel width (see figure below). The moments in both directions are larger in the center portion of the slab than in regions close to the edges. Correspondingly, it is provided that the entire middle strip be designed for the full, tabulated design moment. In the edge strips this
moment is assumed to decrease from its full value at the edge of the middle strip to onethird of this value at the edge of the panel. This distribution is shown for the moments
in
the short span direction in figure below. The lateral variation of the long span moments is similiar.
The discussion so far has been restricted to a single panel simply supported at all four edges.
An actual situation is shown in next figure, in which a system of beams supports a two-way slab. It is seen that some panels, such as
, have two discontinuous exterior edges, while the
other edges are continuous with their neighbors. Panel three continuous edges, the interior panel
has one edge discontinuous and
has all edges continuous, and so on. At a
continuous edge in a slab, moments are negative, just as at interior supports of continuous beams. Also, the magnitude of the positive moments depends on the conditions of continuity at kall four edges. 251
Reinforced Concrete II
Dr. Nasr Abboushi
Correspondingly, Table 1 gives moment coefficients
, for negative moments at continuous
edges. The details of the tables are self-explanatory. Maximum negative edge moments are obtained when both panels adjacent to the particular edge carry full dead and live load. Hence the moment is computed for this total load. Negative moments at discontinuous edges are assumed equal to one-third of the positive moments for the same direction. One must provide for such moments because some degree of restraint is generally provided at discontinuous edges by the torsional rigidity of the edge beam or by the supporting wall. For positive moments there will be little, if any, rotation at the continuous edges if dead load alone is acting, because the loads on both adjacent panels tend to produce opposite rotations which cancel, or nearly so. For this condition, the continuous edges can be regarded as fixed, and the appropriate coefficients for the dead load positive moments are given in Table 2. On the other hand, the maximum live load positive moments are obtained when live load is placed only on the particular panel and not on any of the adjacent panels.
In this case, some rotation will occur at all continuous edges. As an approximation it is assumed that there is
restraint for calculating these live load moments. The
corresponding coefficients are given in Table 3. Finally, for computing shear in the slab and loads on the supporting beams, Table 4 gives the fractions of the total load transmitted in the two directions.
252
that are
Reinforced Concrete II
Dr. Nasr Abboushi
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Reinforced Concrete II
Dr. Nasr Abboushi
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Reinforced Concrete II
Dr. Nasr Abboushi
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Reinforced Concrete II
Dr. Nasr Abboushi
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Reinforced Concrete II
Dr. Nasr Abboushi
Example (Design of two-way edge-supported solid slab ):
A monolithic reinforced concrete floor is to be composed of rectangular bays
measuring
, as shown. Beams of width
and depth
are provided on
all column lines; thus the clear-span dimensions for the two-way slab panels are . The floor is to be designed to carry a service live load
on the slab due to self-weight self -weight plus weight of: •
Tiles,
•
Mortar,
•
Sand,
•
Plaster,
•
Partitions,
.
.
.
.
.
Given: and
and a dead load
.
Find the required slab thickness and
Corner
reinforcement for the corner panel shown.
Panel
Solution:
1. Minimum
thickness
(deflection
requirements): For slabs of this type the first trial
thickness is often taken equal to
Check for the minimum thickness of the slab:
Exterior beam:
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Reinforced Concrete II
Dr. Nasr Abboushi
Interior beam:
Slab section for Exterior beam:
Short direction
Long direction
Slab section for Interior beam:
Short direction
Long direction
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Reinforced Concrete II
Dr. Nasr Abboushi
∑
Corner Panel
the minimum slab thickness will be:
First trial thickness
. Take slab thickness thickness
2. Loads calculation:
Quality Density
Material
Tiles
mortar Sand
Reinforced Concrete solid slab
Plaster
Partitions
Total Dead Load
Dead Load of slab Live Load of slab
,
,
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Reinforced Concrete II
Dr. Nasr Abboushi
3. Moments calculations:
Discontinuous Edge
e g d E s u o u n i t n o c s i D
e g d E s u o u n i t n o C
Case 4
middle strip - short
Column strip - short
Continuous Edge
l
l
l
i
i
i
l
and
l
i
The moment calculations will be done for the slab middle strip.
Negative moments at continuous edges (Table 1):
and
and
Positive moments (Table 2 and Table 3):
and
and
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Column strip - short
l
Reinforced Concrete II
Dr. Nasr Abboushi
and
and
Negative moments at Discontinuous edges (
):
short direction
Moments
n o i t c e r i d g n o l
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Reinforced Concrete II
Dr. Nasr Abboushi
4. Slab reinforcement:
Short direction :
Assume bar diameter
for main reinforcement.
Midspan:
Use
Provide
then
Take
Note that in the edge strips the positive moment, and the corresponding steel reinforcement
area, is assumed to decrease from its full value at the edge of the middle strip to one -third of this value at the edge of the panel, which will not be provided.
Continuous edge:
Assume bar diameter
for main reinforcement.
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Reinforced Concrete II
Dr. Nasr Abboushi
Use
Provide
then
Take
Discontinuous edge.
The negative moment at the discontinuous edge is one-third the positive moment in the
span.
Provide
Take
Long direction.
Design for positive and negative moment as in the short direction. Note that the effective depth for the long direction will be
5. Check for shear:
and
and
The reactions of the slab are calculated from Table 4, which indicates that transmitted in the short direction and
in the long direction.
The total load on the panel being
The load per meter on the long beam is
of the load is
, and
The load per meter on the short beam is
.
The shear to be transmitted by the slab to these beams is numerically equal to these beam loads, reduced to a critical section a distance
from the beam face. The shear strength of
the slab is
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Reinforced Concrete II
Dr. Nasr Abboushi
√ - for shear.
The thickness of the slab is adequate enough. at the face of support,
at distance from the face of support will be
smaller.
Even, if
for solid slabs, the thickness of the slab will be enough .
Example:
A monolithic reinforced concrete solid slab is to be composed of rectangular bays as shown. Beams of width
and depth
are provided on all column lines. The floor is to be
designed to carry a service live load
and a dead load on the slab due to self-
weight plus weight of: • • • • •
Tiles,
.
Mortar, Sand,
Plaster,
.
.
.
Partitions,
Given:
.
and
.
Find the required slab thickness and reinforcement for the solid slab shown. Solution:
1. Minimum thickness (deflection requirements):
For slabs of this type the first trial thickness is often taken equal to
Take
As was required in the previous example. As
Check for the minimum thickness of the slab (for
shaded panel):
Exterior beam:
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Dr. Nasr Abboushi
6. 0 m
6.0 m
B 10 (60 x 60 cm)
B 11 (60 x 60 cm)
m 1 . 8
) m c 0 6 x 0 6 ( 5 1 B
m 1 . 8
)
m c 0 6 x 0 6
( 8 1 B
( 1 2 B
B 7 (60 x 60 cm)
B 8 (60 x 60 cm)
B 9 (60 x 60 cm)
7 1 B
6. 6 m
) m c 0 6 x 0 6 ( 6 1 B
) m c 0 6 x 0 ( 6 0 2 B
B 5 (60 x 60 cm)
B 6 (60 x 60 cm)
B 1 (60 x 60 cm)
)
m c 0 6 x 0 6 (
B 2 (60 x 60 cm)
6.6 m
265
) m c 0 6 x 0 6 ( 4 2 B
m 5 . 7
)
m c 0 6 x 0 6
B 4 (60 x 60 cm)
m 6 . 8
) m c 0 6 x 0 6 ( 3 1 B
B 12 (60 x 60 cm)
) m c 0 6 x 0 6 ( 4 1 B
6. 0 m
)
m c 0 6 x 0 6 (
9 1 B
B 3 (60 x 60 cm)
6. 6 m
) m c 0 6 x 0 6 ( 3 2 B
) m c 0 6 x 0 6 ( 2 2 B
m 5 . 7
m 0 . 8
Reinforced Concrete II
Dr. Nasr Abboushi
Interior beam:
Slab section for Exterior beam:
Short direction
Long direction
Slab section for Interior beam:
Short direction
Long direction
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Reinforced Concrete II
Dr. Nasr Abboushi
∑
Corner Panel
the minimum slab thickness will be:
First trial thickness
. Take slab thickness
2. Loads calculation:
Quality Density
Material
Tiles
mortar Sand
Reinforced Concrete solid slab
Plaster
Partitions
Total Dead Load
Dead Load of slab Live Load of slab
,
,
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