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Case Study #1 • Under what circumstances, the flat plate/slab or ribbed slab system are applicable ? • What are the merits and demerits if they are adopted?

Primary Structure Floor system -material: composite structural steel/concrete - type: beams, composite metal deck/concrete floor - pattern: radial beams, edge beams and one-way floor slab - beam clear span 14 m - floor slab span 2.2 - 3.25m

Core - material reinforced concrete

Note: - If the floor plan is rectangular, ribbed slab system could have been adopted.

Lecture Goals Design of Two-Way Floor Slab System (including ribbed and waffle slab system)

• One-way and two-way slab • Slab thickness, h • Simplified Coefficient Method

Comparison of One-way and Two-way slab behavior One-way slabs carry load in one direction.

Comparison of One-way and Two-way slab behavior One-way and two-way slab action carry load in two directions.

Two-way slabs carry load in two directions.

One-way slabs: Generally, long side/short side > 1.5

Comparison of One-way and Two-way slab behavior

Comparison between a two-way slab verses a one-way slab For flat plates and slabs the column connections can vary between:

Flat slab

Two-way slab with beams

Comparison of One-way and Two-way slab behavior

Flat Plate

Waffle slab

Comparison of One-way and Two-way slab behavior Economic Choices • Flat Plate suitable span 6 to 7.5m with LL= 3 ~ 5 kN/m2 Advantages – Low cost formwork – Exposed flat ceilings – Fast Disadvantages – Low shear capacity – Low Stiffness (notable deflection)

Comparison of One-way and Two-way slab behavior

The two-way ribbed slab and waffled slab system: General thickness of the slab is 50 to 100 mm.

Comparison of One-way and Two-way slab behavior Economic Choices • Flat Slab suitable span 6 to 9m with LL= 4 ~ 7.5 kN/m2 Advantages – Low cost formwork – Exposed flat ceilings – Fast Disadvantages – Need more formwork for capital and panels

Comparison of One-way and Two-way slab behavior Economic Choices • Waffle Slab suitable span 9 to 15m with LL= 4 ~ 7.5 kN/m2 Advantages – Carries heavy loads – Attractive exposed ceilings – Fast Disadvantages – Formwork with panels is expensive

Comparison of One-way and Two-way slab behavior

Comparison of One-way and Two-way slab behavior Economic Choices • One-way Slab on beams suitable span 3 to 6m with LL= 3 ~ 5 kN/m2 – Can be used for larger spans with relatively higher cost and higher deflections • One-way joist floor system is suitable span 6 to 9m with LL= 4 ~ 6 kN/m2 – Deep ribs, the concrete and steel quantities are relative low – Expensive formwork expected.

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

ws =load taken by short direction wl = load taken by long direction δA = δB 5ws A4 384 EI

ws wl

=

B4 A

4

=

5wl B 4 384 EI

For B = 2A ⇒ ws = 16wl

Rule of Thumb: For B/A > 2, design as one-way slab

Analogy of two-way slab to plank and beam floor Section A-A: 2

Moment per m width in planks ⇒ M = wl1 kNm per m 8 Total Moment

⇒ M f = (wl 2 )

l12 kNm 8

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

Analogy of two-way slab to plank and beam floor wl Uniform load on each beam ⇒ 1 kN/m 2 2 ⎛ wl ⎞ l ⇒ Mlb = ⎜ 1 ⎟ 2 kN Moment in one beam (Sec: B-B) ⎝ 2 ⎠8

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

l 22 kNm 8 Full load was transferred east-west by the planks and then was transferred north-south by the beams;

Total Moment in both beams

⇒ M = (wl1 )

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

General Design Concepts (1) Simplified Coefficient Method (SCM) Limited to slab systems to uniformly distributed loads and supported on equally spaced columns. Method uses a set of coefficients to determine the design moment at critical sections. Two-way slab system that do not meet the limitations of the ACI Code 13.6.1 or BS8110 must be analyzed more accurate procedures.

General Design Concepts (2) Equivalent Frame Method (EFM) A three dimensional 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.

Equivalent Frame Method (EFM)

Longitudinal equivalent frame

Transverse equivalent frame

Equivalent Frame Method (EFM)

Elevation of the frame

Perspective view

Method of Analysis Method of Analysis (1) Elastic Analysis Concrete slab may be treated as an elastic plate. Use Timoshenko’s method of analyzing the structure. Finite element analysis

(2) Plastic Analysis The yield method used to determine the limit state of slab by considering the yield lines that occur in the slab as a collapse mechanism. The strip method, where slab is divided into strips and the load on the slab is distributed in two orthogonal directions and the strips are analyzed as beams. The optimal analysis presents methods for minimizing the reinforcement based on plastic analysis

Method of Analysis (3) Nonlinear analysis Simulates the true load-deformation characteristics of a reinforced concrete slab with finite-element method takes into consideration of nonlinearities of the stress-strain relationship of the individual members.

Basic Steps in Two-way Slab Design 1. Choose layout and type of slab. 2. Choose slab thickness to control deflection. Also, check if thickness is adequate for shear. 3. 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

Simplified Coefficient Method

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

Minimum Slab Thickness for two-way construction

Minimum Thickness, h (only available in ACI code)

Minimum Slab Thickness for two-way construction The definitions of the terms are: h = Minimum slab thickness without interior beams ln = Clear span in the long direction measured face to face of column β = the ratio of the long to short clear span αm= The average value of α for all beams on the sides of the panel.

The ACI Code 9.5.3 specifies a minimum slab thickness to control deflection. There are three empirical limitations for calculating the slab thickness (h), which are based on experimental research. If these limitations are not met, it will be necessary to compute deflection.

Definition of Beam-to-Slab Stiffness Ratio, α Accounts for stiffness effect of beams located along slab edge reduces deflections of panel adjacent to beams.

α=

flexural stiffness of beam flexural stiffness of slab

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

4E cb I b / l 4E cs I s / l

=

Beam and Slab Sections for calculation of α

E cb I b E cs I s

E cb = Modulus of elasticity of beam concrete E sb = Modulus of elasticity of slab concrete I b = Moment of inertia of uncracked beam I s = Moment of inertia of uncracked slab

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

Beam and Slab Sections for calculation of α

Beam and Slab Sections for calculation of α

Definition of beam cross-section Charts may be used to calculate α Fig. 13-21

Minimum Slab Thickness for two-way construction (a) For 0.2 ≤ α m ≤ 2

f ⎞ ⎛ l n ⎜⎜ 0.8 + y ⎟⎟ 1400 ⎠ ⎝ h= 36 + 5 β (α m − 0.2)

Minimum Slab Thickness for two-way construction (b) For 2 < α m

f ⎞ ⎛ l n ⎜⎜ 0.8 + y ⎟⎟ 1400 ⎠ h= ⎝ 36 + 9 β

fy in MPa. But not less than 125 mm.

fy in MPa. But not less than 88 mm.

Minimum Slab Thickness for two-way construction

Minimum Slab Thickness for two-way construction

(c) For α m < 0.2 Use the following table

Slabs without interior beams spanning between supports and ratio of long span to short span < 2 See section 9.5.3.3 For slabs with beams spanning between supports on all sides.

270 MPa 413 MPa 517 MPa

Minimum Slab Thickness for two-way construction Slabs without drop panels meeting 13.3.7.1 and 13.3.7.2, tmin = 125 mm Slabs with drop panels meeting 13.3.7.1 and 13.3.7.2, tmin = 100 mm

Example A flat plate floor system with panels 7.2m by 6m is supported on 500mm square columns. Determine the minimum slab thickness required for the interior and corner panels. Use fcu = 30 MPa and fy = 460 MPa.

Example Example The floor system consists of solid slabs and beams in two directions supported on 500mm square columns. Determine the minimum slab thickness required for an interior panel. Use fcu = 30 MPa and fy = 460 MPa.

The cross-sections are:

Example The resulting cross section:

Minimum Slab Thickness for two-way construction Maximum Spacing of Reinforcement s ≤ 2t (ACI 13.3.2 ) At points of max. +/- M: s ≤ 2d (BS 8110 )

and s ≤ 450 mm. (ACI 7.12.3 )

Max. and Min Reinforcement Requirements

As (min ) = As (T &S ) from ACI 7.12 (ACI 13.3.1) As (max ) = 0.75 As (bal )

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

Simplified Method for Two-way Slab Limitations on use of Simplified Coefficient method 3. Successive span in each direction shall not differ by more than 1/3 the longer span. 4. Columns may be offset from the basic rectangular grid of the building by up to 0.1 times the span parallel to the offset.

Simplified Method for Two-way Slab Limitations on use of Simplified Coefficient method 5. All loads must be due to gravity only (N/A to unbraced laterally loaded frames, from mats or pre-stressed slabs) 6. Service (unfactored) live load ≤ 2 service dead load

Simplified Method for Two-way Slab Limitations on use of Simplified method 7. For panels with beams between supports on all sides, relative stiffness of the beams in the 2 perpendicular directions.

α1l22

α 2l12 Shall not be less than 0.2 nor greater than 5.0

Definition of Beam-to-Slab Stiffness Ratio, α Accounts for stiffness effect of beams located along slab edge reduces deflections of panel adjacent to beams.

α=

flexural stiffness of beam flexural stiffness of slab

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

4E cb I b / l 4E cs I s / l

=

4E cb I b 4E cs I s

E cb = Modulus of elasticity of beam concrete E sb = Modulus of elasticity of slab concrete I b = Moment of inertia of uncracked beam I s = Moment of inertia of uncracked slab

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

Beam and Slab Sections for calculation of α

Beam and Slab Sections for calculation of α

Beam and Slab Sections for calculation of α

Column and Middle Strips The slab is broken up into column and middle strips for analysis

Definition of beam cross-section Charts may be used to calculate α Fig. 13-21

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

Distribution of Moments

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

Column Strips and Middle Strips

Total static Moment, Mo

M0 =

wu l2ln2

(ACI 13 - 3)

8 where w = design load per unit area u l 2 = transverse width of the strip l n = clear span between columns

(for circular columns, calc. ln using h = 0.886dc )

Moments vary across width of slab panel

∴

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

Column Strips and Middle Strips Column strips Design w/width on either side of a column centerline equal to smaller of ⎧0.25 l2 ⎨ ⎩ 0.25 l1

Column Strips and Middle Strips Middle strips: Design strip bounded by two column strips.

l1= length of span in direction moments are being determined. l2= length of span transverse to l1

Positive and Negative Moments in Panels Moment coefficients are assigned to + M and -M Rules given in BS8110 Table 3.12

Shear in Panels Shear coefficients are given in BS8110 Table 3.12

0.6 F +0.063 FL

-0.086 FL

0.5 F

F is the total design load on the strip of slab between adjacent columns. -0.063 FL

Distribution of moment in 2-way slab without beam (flat slab) Column Middle strip strip -M

75%

25%

+M

55%

45%

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Primary Structure Floor system -material: composite structural steel/concrete - type: beams, composite metal deck/concrete floor - pattern: radial beams, edge beams and one-way floor slab - beam clear span 14 m - floor slab span 2.2 - 3.25m

Core - material reinforced concrete

Note: - If the floor plan is rectangular, ribbed slab system could have been adopted.

Lecture Goals Design of Two-Way Floor Slab System (including ribbed and waffle slab system)

• One-way and two-way slab • Slab thickness, h • Simplified Coefficient Method

Comparison of One-way and Two-way slab behavior One-way slabs carry load in one direction.

Comparison of One-way and Two-way slab behavior One-way and two-way slab action carry load in two directions.

Two-way slabs carry load in two directions.

One-way slabs: Generally, long side/short side > 1.5

Comparison of One-way and Two-way slab behavior

Comparison between a two-way slab verses a one-way slab For flat plates and slabs the column connections can vary between:

Flat slab

Two-way slab with beams

Comparison of One-way and Two-way slab behavior

Flat Plate

Waffle slab

Comparison of One-way and Two-way slab behavior Economic Choices • Flat Plate suitable span 6 to 7.5m with LL= 3 ~ 5 kN/m2 Advantages – Low cost formwork – Exposed flat ceilings – Fast Disadvantages – Low shear capacity – Low Stiffness (notable deflection)

Comparison of One-way and Two-way slab behavior

The two-way ribbed slab and waffled slab system: General thickness of the slab is 50 to 100 mm.

Comparison of One-way and Two-way slab behavior Economic Choices • Flat Slab suitable span 6 to 9m with LL= 4 ~ 7.5 kN/m2 Advantages – Low cost formwork – Exposed flat ceilings – Fast Disadvantages – Need more formwork for capital and panels

Comparison of One-way and Two-way slab behavior Economic Choices • Waffle Slab suitable span 9 to 15m with LL= 4 ~ 7.5 kN/m2 Advantages – Carries heavy loads – Attractive exposed ceilings – Fast Disadvantages – Formwork with panels is expensive

Comparison of One-way and Two-way slab behavior

Comparison of One-way and Two-way slab behavior Economic Choices • One-way Slab on beams suitable span 3 to 6m with LL= 3 ~ 5 kN/m2 – Can be used for larger spans with relatively higher cost and higher deflections • One-way joist floor system is suitable span 6 to 9m with LL= 4 ~ 6 kN/m2 – Deep ribs, the concrete and steel quantities are relative low – Expensive formwork expected.

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

ws =load taken by short direction wl = load taken by long direction δA = δB 5ws A4 384 EI

ws wl

=

B4 A

4

=

5wl B 4 384 EI

For B = 2A ⇒ ws = 16wl

Rule of Thumb: For B/A > 2, design as one-way slab

Analogy of two-way slab to plank and beam floor Section A-A: 2

Moment per m width in planks ⇒ M = wl1 kNm per m 8 Total Moment

⇒ M f = (wl 2 )

l12 kNm 8

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

Analogy of two-way slab to plank and beam floor wl Uniform load on each beam ⇒ 1 kN/m 2 2 ⎛ wl ⎞ l ⇒ Mlb = ⎜ 1 ⎟ 2 kN Moment in one beam (Sec: B-B) ⎝ 2 ⎠8

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

l 22 kNm 8 Full load was transferred east-west by the planks and then was transferred north-south by the beams;

Total Moment in both beams

⇒ M = (wl1 )

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

General Design Concepts (1) Simplified Coefficient Method (SCM) Limited to slab systems to uniformly distributed loads and supported on equally spaced columns. Method uses a set of coefficients to determine the design moment at critical sections. Two-way slab system that do not meet the limitations of the ACI Code 13.6.1 or BS8110 must be analyzed more accurate procedures.

General Design Concepts (2) Equivalent Frame Method (EFM) A three dimensional 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.

Equivalent Frame Method (EFM)

Longitudinal equivalent frame

Transverse equivalent frame

Equivalent Frame Method (EFM)

Elevation of the frame

Perspective view

Method of Analysis Method of Analysis (1) Elastic Analysis Concrete slab may be treated as an elastic plate. Use Timoshenko’s method of analyzing the structure. Finite element analysis

(2) Plastic Analysis The yield method used to determine the limit state of slab by considering the yield lines that occur in the slab as a collapse mechanism. The strip method, where slab is divided into strips and the load on the slab is distributed in two orthogonal directions and the strips are analyzed as beams. The optimal analysis presents methods for minimizing the reinforcement based on plastic analysis

Method of Analysis (3) Nonlinear analysis Simulates the true load-deformation characteristics of a reinforced concrete slab with finite-element method takes into consideration of nonlinearities of the stress-strain relationship of the individual members.

Basic Steps in Two-way Slab Design 1. Choose layout and type of slab. 2. Choose slab thickness to control deflection. Also, check if thickness is adequate for shear. 3. 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

Simplified Coefficient Method

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

Minimum Slab Thickness for two-way construction

Minimum Thickness, h (only available in ACI code)

Minimum Slab Thickness for two-way construction The definitions of the terms are: h = Minimum slab thickness without interior beams ln = Clear span in the long direction measured face to face of column β = the ratio of the long to short clear span αm= The average value of α for all beams on the sides of the panel.

The ACI Code 9.5.3 specifies a minimum slab thickness to control deflection. There are three empirical limitations for calculating the slab thickness (h), which are based on experimental research. If these limitations are not met, it will be necessary to compute deflection.

Definition of Beam-to-Slab Stiffness Ratio, α Accounts for stiffness effect of beams located along slab edge reduces deflections of panel adjacent to beams.

α=

flexural stiffness of beam flexural stiffness of slab

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

4E cb I b / l 4E cs I s / l

=

Beam and Slab Sections for calculation of α

E cb I b E cs I s

E cb = Modulus of elasticity of beam concrete E sb = Modulus of elasticity of slab concrete I b = Moment of inertia of uncracked beam I s = Moment of inertia of uncracked slab

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

Beam and Slab Sections for calculation of α

Beam and Slab Sections for calculation of α

Definition of beam cross-section Charts may be used to calculate α Fig. 13-21

Minimum Slab Thickness for two-way construction (a) For 0.2 ≤ α m ≤ 2

f ⎞ ⎛ l n ⎜⎜ 0.8 + y ⎟⎟ 1400 ⎠ ⎝ h= 36 + 5 β (α m − 0.2)

Minimum Slab Thickness for two-way construction (b) For 2 < α m

f ⎞ ⎛ l n ⎜⎜ 0.8 + y ⎟⎟ 1400 ⎠ h= ⎝ 36 + 9 β

fy in MPa. But not less than 125 mm.

fy in MPa. But not less than 88 mm.

Minimum Slab Thickness for two-way construction

Minimum Slab Thickness for two-way construction

(c) For α m < 0.2 Use the following table

Slabs without interior beams spanning between supports and ratio of long span to short span < 2 See section 9.5.3.3 For slabs with beams spanning between supports on all sides.

270 MPa 413 MPa 517 MPa

Minimum Slab Thickness for two-way construction Slabs without drop panels meeting 13.3.7.1 and 13.3.7.2, tmin = 125 mm Slabs with drop panels meeting 13.3.7.1 and 13.3.7.2, tmin = 100 mm

Example A flat plate floor system with panels 7.2m by 6m is supported on 500mm square columns. Determine the minimum slab thickness required for the interior and corner panels. Use fcu = 30 MPa and fy = 460 MPa.

Example Example The floor system consists of solid slabs and beams in two directions supported on 500mm square columns. Determine the minimum slab thickness required for an interior panel. Use fcu = 30 MPa and fy = 460 MPa.

The cross-sections are:

Example The resulting cross section:

Minimum Slab Thickness for two-way construction Maximum Spacing of Reinforcement s ≤ 2t (ACI 13.3.2 ) At points of max. +/- M: s ≤ 2d (BS 8110 )

and s ≤ 450 mm. (ACI 7.12.3 )

Max. and Min Reinforcement Requirements

As (min ) = As (T &S ) from ACI 7.12 (ACI 13.3.1) As (max ) = 0.75 As (bal )

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

Simplified Method for Two-way Slab Limitations on use of Simplified Coefficient method 3. Successive span in each direction shall not differ by more than 1/3 the longer span. 4. Columns may be offset from the basic rectangular grid of the building by up to 0.1 times the span parallel to the offset.

Simplified Method for Two-way Slab Limitations on use of Simplified Coefficient method 5. All loads must be due to gravity only (N/A to unbraced laterally loaded frames, from mats or pre-stressed slabs) 6. Service (unfactored) live load ≤ 2 service dead load

Simplified Method for Two-way Slab Limitations on use of Simplified method 7. For panels with beams between supports on all sides, relative stiffness of the beams in the 2 perpendicular directions.

α1l22

α 2l12 Shall not be less than 0.2 nor greater than 5.0

Definition of Beam-to-Slab Stiffness Ratio, α Accounts for stiffness effect of beams located along slab edge reduces deflections of panel adjacent to beams.

α=

flexural stiffness of beam flexural stiffness of slab

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

4E cb I b / l 4E cs I s / l

=

4E cb I b 4E cs I s

E cb = Modulus of elasticity of beam concrete E sb = Modulus of elasticity of slab concrete I b = Moment of inertia of uncracked beam I s = Moment of inertia of uncracked slab

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

Beam and Slab Sections for calculation of α

Beam and Slab Sections for calculation of α

Beam and Slab Sections for calculation of α

Column and Middle Strips The slab is broken up into column and middle strips for analysis

Definition of beam cross-section Charts may be used to calculate α Fig. 13-21

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

Distribution of Moments

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

Column Strips and Middle Strips

Total static Moment, Mo

M0 =

wu l2ln2

(ACI 13 - 3)

8 where w = design load per unit area u l 2 = transverse width of the strip l n = clear span between columns

(for circular columns, calc. ln using h = 0.886dc )

Moments vary across width of slab panel

∴

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

Column Strips and Middle Strips Column strips Design w/width on either side of a column centerline equal to smaller of ⎧0.25 l2 ⎨ ⎩ 0.25 l1

Column Strips and Middle Strips Middle strips: Design strip bounded by two column strips.

l1= length of span in direction moments are being determined. l2= length of span transverse to l1

Positive and Negative Moments in Panels Moment coefficients are assigned to + M and -M Rules given in BS8110 Table 3.12

Shear in Panels Shear coefficients are given in BS8110 Table 3.12

0.6 F +0.063 FL

-0.086 FL

0.5 F

F is the total design load on the strip of slab between adjacent columns. -0.063 FL

Distribution of moment in 2-way slab without beam (flat slab) Column Middle strip strip -M

75%

25%

+M

55%

45%

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