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**Reinforced Concrete Design-8**

Design of 2 way Slabs By Dr. Attaullah Shah Swedish College of Engineering and Technology Wah Cantt.

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**One way vs Two way slab system**

L>2S As ≥ Temp. steel Min. Spacing ≥ ∅ main steel ≥ 4/3 max agg. ≥ 2.5 cm (1in) Max. Spacing ≤ 3 t ≤ 45 cm (17in) Min slab thickness =

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**Shear Strength of Slabs**

In two-way floor systems, the slab must have adequate thickness to resist both bending moments and shear forces at critical section. There are three cases to look at for shear. 1. 2. 3. Two-way Slabs supported on beams Two-Way Slabs without beams Shear Reinforcement in two-way slabs without beams.

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**Shear Strength of Slabs**

Two-way slabs supported on beams The critical location is found at d distance from the column, where The supporting beams are stiff and are capable of transmitting floor loads to the columns.

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**Shear Strength of Slabs**

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

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**Shear Strength of Slabs**

Two-Way Slabs without beams There are two types of shear that need to be addressed 1. 2. One-way shear or beam shear at distance d from the column Two-way or punch out shear which occurs along a truncated cone.

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**Shear Strength of Slabs**

One-way shear or beam shear at distance d from the column Two-way or punch out shear which occurs along a truncated cone. 1. 2.

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**Shear Strength of Slabs**

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

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**Shear Strength of Slabs**

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

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**Shear Strength of Slabs**

If shear reinforcement is not provided, the shear strength of concrete is the smaller of: perimeter of the critical section ratio of long side of column to short side bo = bc =

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**Shear Strength of Slabs**

If shear reinforcement is not provided, the shear strength of concrete is the smaller of: as is 40 for interior columns, 30 for edge columns, and 20 for corner columns.

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**Shear Strength of Slabs**

Shear Reinforcement in two-way slabs without beams. For plates and flat slabs, which do not meet the condition for shear, one can either - Increase slab thickness - Add reinforcement Reinforcement can be done by shearheads, anchor bars, conventional stirrup cages and studded steel strips.

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**Shear Strength of Slabs**

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

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**Shear Strength of Slabs**

Anchor bars consists of steel reinforcement rods or bent bar reinforcement

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**Shear Strength of Slabs**

Conventional stirrup cages

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**Shear Strength of Slabs**

Studded steel strips

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**Shear Strength of Slabs**

The reinforced slab follows section in the ACI Code, where Vn can not The spacing, s, can not exceed d/2. If a shearhead reinforcement is provided

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

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Example Problem Compute the shear terms find b0 for

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**Example Problem Compute the maximum allowable shear**

Vu =195 k > k Shear reinforcement is need!

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**Example Problem Compute the maximum allowable shear**

So fVn >Vu Can use shear reinforcement

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

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**Example Problem Determine the a for The depth = a+d**

= 41.8 in in. = 49.3 in. 50 in.

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**Example Problem Determine shear reinforcement**

The fVs per side is fVs / 4 = k

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**Example Problem Determine shear reinforcement**

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

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**Example Problem Determine shear reinforcement spacing**

Maximum allowable spacing is

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**Example Problem Use s = 3.5 in.**

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

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**Example Problem The final result:**

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

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

Method of dividing total static moment Mo into positive and negative moments. Limitations on use of Direct Design method 1. 2. Minimum of 3 continuous spans in each direction. (3 x 3 panel) Rectangular panels with long span/short span

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

Limitations on use of Direct Design method 3. 4. Successive span in each direction shall not differ by more than 1/3 the longer span. Columns may be offset from the basic rectangular grid of the building by up to 0.1 times the span parallel to the offset.

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

Limitations on use of Direct Design method 5. 6. All loads must be due to gravity only (N/A to unbraced laterally loaded frames, from mats or pre-stressed slabs) Service (unfactored) live load service dead load

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

Limitations on use of Direct Design method 7. For panels with beams between supports on all sides, relative stiffness of the beams in the 2 perpendicular directions. Shall not be less than 0.2 nor greater than 5.0

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

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

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

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

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**Two-Way Slab Design Static Equilibrium of Two-Way Slabs**

Analogy of two-way slab to plank and beam floor Section A-A: Moment per ft width in planks Total Moment

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**Two-Way Slab Design Static Equilibrium of Two-Way Slabs**

Analogy of two-way slab to plank and beam floor Uniform load on each beam Moment in one beam (Sec: B-B)

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**Two-Way Slab Design Static Equilibrium of Two-Way Slabs**

Total Moment in both beams Full load was transferred east-west by the planks and then was transferred north-south by the beams; The same is true for a two-way slab or any other floor system.

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

1. 2. 3. Choose layout and type of slab. Choose slab thickness to control deflection. Also, check if thickness is adequate for shear. Choose Design method Equivalent Frame Method- use elastic frame analysis to compute positive and negative moments Direct Design Method - uses coefficients to compute positive and negative slab moments

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

4. 5. 6. 7. 8. Calculate positive and negative moments in the slab. Determine distribution of moments across the width of the slab. - Based on geometry and beam stiffness. Assign a portion of moment to beams, if present. Design reinforcement for moments from steps 5 and 6. Check shear strengths at the columns

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

Maximum Spacing of Reinforcement At points of max. +/- M: Min Reinforcement Requirements

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**Distribution of Moments**

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

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**Distribution of Moments**

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

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**Distribution of Moments**

Total static Moment, Mo where

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**Column Strips and Middle Strips**

Moments vary across width of slab panel Design moments are averaged over the width of column strips over the columns & middle strips between column strips.

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**Column Strips and Middle Strips**

Column strips Design w/width on either side of a column centerline equal to smaller of l1= length of span in direction moments are being determined. l2= length of span transverse to l1

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**Column Strips and Middle Strips**

Middle strips: Design strip bounded by two column strips.

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**Positive and Negative Moments in Panels**

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

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Moment Distribution

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**Positive and Negative Moments in Panels**

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

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**Longitudinal Distribution of Moments in Slabs**

For a typical interior panel, the total static moment is divided into positive moment 0.35 Mo and negative moment of Mo. For an exterior panel, the total static moment is dependent on the type of reinforcement at the outside edge.

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Distribution of M0

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Moment Distribution The factored components of the moment for the beam.

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**Transverse Distribution of Moments**

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

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**Transverse Distribution of Moments**

Transverse distribution of the longitudinal moments to middle and column strips is a function of the ratio of length l2/l1,a1, and bt.

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**Transverse Distribution of Moments**

Transverse distribution of the longitudinal moments to middle and column strips is a function of the ratio of length l2/l1,a1, and bt. torsional constant

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**Distribution of M0 ACI Sec 13.6.3.4**

For spans framing into a common support negative moment sections shall be designed to resist the larger of the 2 interior Mu’s ACI Sec Edge beams or edges of slab shall be proportioned to resist in torsion their share of exterior negative factored moments

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**Factored Moment in Column Strip**

Ratio of flexural stiffness of beam to stiffness of slab in direction l1. Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length) bt=

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**Factored Moment in an Interior Strip**

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**Factored Moment in an Exterior Panel**

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**Factored Moment in an Exterior Panel**

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**Factored Moment in Column Strip**

Ratio of flexural stiffness of beam to stiffness of slab in direction l1. Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length) bt=

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**Factored Moment in Column Strip**

Ratio of flexural stiffness of beam to stiffness of slab in direction l1. Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length) bt=

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**Factored Moment in Column Strip**

Ratio of flexural stiffness of beam to stiffness of slab in direction l1. Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length) bt=

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**Factored Moments Factored Moments in beams (ACI Sec. 13.6.3)**

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

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**Factored Moments Factored Moments in Middle strips (ACI Sec. 13.6.3)**

The portion of the + Mu and - Mu not resisted by column strips shall be proportionately assigned to corresponding half middle strips. Each middle strip shall be proportioned to resist the sum of the moments assigned to its 2 half middle strips.

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**ACI Provisions for Effects of Pattern Loads**

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

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**ACI Provisions for Effects of Pattern Loads**

1. 2. 3. The ratio of live to dead load. A high ratio will increase the effect of pattern loadings. The ratio of column to beam stiffness. A low ratio will increase the effect of pattern loadings. Pattern loadings. Maximum positive moments within the spans are less affected by pattern loadings.

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**Reinforcement Details Loads**

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

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**Reinforcement Details Loads**

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

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**Minimum extension for reinforcement in slabs without beams(Fig. 13. 3**

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Moment Distribution The factored components of the moment for the beam.

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**Transverse Distribution of Moments**

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

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**Limitations of Direct Design Method**

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ANALYSIS OF TWO-WAY SLABS WITH BEAMS

ANALYSIS OF TWO-WAY SLABS WITH BEAMS

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