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Introduction When a rectangular slab is supported on all the sides and the length-to-breadth ratio is less than two, it is considered to be a two-way slab. The slab spans in both the orthogonal directions. A circular slab is a two-way slab. In general, a slab which is not falling in the category of one-way slab, is considered to be a two-way slab. Rectangular two-way slabs can be divided into the following types. 1) Flat plates: These slabs do not have beams between the columns, drop panels or column capitals. Usually, there are spandrel beams at the edges. 2) Flat slabs: These slabs do not have beams but have drop panels or column capitals. 3) Two-way slabs with beams: There are beams between the columns. If the beams are wide and shallow, they are termed as band beams. For long span construction, there are ribs in both the spanning directions of the slab. This type of slabs is called waffle slabs. The slabs can be cast-in-situ (cast-in-place). Else, the slabs can be precast at ground level and lifted to the final height. The later type of slabs is called lift slabs. A slab in a framed building can be a two-way slab depending upon its length-to-breadth (L / B) ratio. Two-way slabs are also present as mat (raft) foundation. The following sketches show the plan of various cases of two-way slabs. The spanning directions in each case are shown by the double headed arrows. `

(a) Two way beam supported slab

(c) Two way beam supFlatported slab

(b) Flat Plate

(d) Waffle Slab

Types of Two-Way Slab Systems

Figure 2 Plans of two-way slabs The absence of beams in flat plates and flat slabs lead to the following advantages. 1) Formwork is simpler 2) Reduced obstruction to service conduits 3) More flexibility in interior layout and future refurbishment. Two-way slabs can be post-tensioned. The main advantage of prestressing a slab is the increased spanto-depth ratio. As per ACI 318-02 (Building Code Requirements for Structural Concrete, American Concrete Institute), the limits of span-to-depth ratios are as follows. For floors 42 For roofs 48.

The values can be increased to 48 and 52, respectively, if the deflection, camber and vibration are not objectionable. The following photographs show post-tensioned flat plate and flat slab.

Analysis The analysis of two-way slabs is given in Section 31, IS:456 - 2000, under “Flat Slabs”.The analysis is applicable to flat plates, flat slabs and two-way slabs with deflecting beams. For two-way slabs with beams, if the beams are sufficiently stiff, then the method (based on moment coefficients) given in Annex D, IS:456 – 2000, is applicable. The direct design method of analysing a two-way slab is not recommended for prestressed slabs. The equivalent frame method is recommended by ACI 318-02. It is given in Subsection 31.5, IS:456 - 2000. This method is briefly covered in this section for flat plates and flat slabs. Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas

(a) Flat plate

(b) Flat slab Figure 3 Post-tensioned two-way slabs (Courtesy: VSL India Pvt. Ltd.)

Flat Slabs A reinforced concrete flat slab, also called as beamless slab, is a slab supported directly by columns without beams. A part of the slab bounded on each of the four sides by centre line of column is called panel. The flat slab is often thickened closed to supporting columns to provide adequate strength in shear and to reduce the amount of negative reinforcement in the support regions. The thickened portion i.e. the projection below the slab is called drop or drop panel. In some cases, the section of column at top, as it meets the floor slab or a drop panel, is enlarged so as to increase primarily the perimeter of the critical section, for shear and hence, increasing the capacity of the slab for resisting two-way shear and to reduce negative bending moment at the support. Such enlarged or flared portion of and a capital. Slabs of constant thickness which do not have drop panels or column capitals are referred to as flat plates. The strength of the flat plate structure is often limited due to punching shear action around columns, and consequently they are used for light loads and relatively small spans In general, in column-supported slabs, with or without beams along the column lines, 100 percent of the slab load has to be transmitted by the floor system in both directions (transverse and longitudinal) towards the columns. In such cases, the entire floor system and the columns act integrally in a two-way frame action. Proportioning of Slab Thickness, Drop Panel and Column Head Slab Thickness The thickness of the slab is generally governed by deflection control criteria. [Shear is also an important design criterion — especially in flat plates (slabs without beams and drop panels) and at exterior column supports]. The calculation of deflections of two-way slab systems is quite complex, and recourse is often made to empirical rules which limit maximum span/depth ratios as indirect measures of deflection control. For this purpose, the Code (Cl. 31.2.1) recommends the same l/d ratios prescribed (in Cl. 23.2, also refer Section 5.3.2) for flexural members in general, with the following important differences:

the longer span should be considered‡ (unlike the case of slabs supported on walls or stiff beams, where the shorter span is considered); for the purpose of calculating the modification factor kt [Table 5.2] for tension reinforcement, an average percentage of steel across the whole width of panel should be considered [Ref. 11.11]; When drop panels conforming to Cl. 31.2.2 are not provided around the column supports, in flat slabs the calculated l/d ratios should be further reduced by a factor of 0.9; the minimum thickness of the flat slab should be 125 mm

Slab Thickness Recommended by other Codes Other empirical equations fo um span/depth ratios have been established,based on the results of extensive tests on floor slabs, and have been supported by past experience with such construction under normal values of uniform loading[Ref. 11.18, 11.19]. Thus Ref. 11.8 recommends equations 11.25 to 11.26a for the minimum overall thickness of slabs necessary for the control of deflections. If these minimum thickness requirements are satisfied, deflections need not be computed. These equations are also applicable for two-way slabs supported on stiff beams. However, these thicknesses may not be the most economical in all cases, and may even be inadequate for slabs with large live to dead load ratios. In the calculation of span/depth ratio, the clear span ln in the longer direction and the overall depth (thickness) D are to be considered. For flat plates and slabs with column capitals, the minimum overall thickness of slab is: D ≥ [ln (0.6 + fy / 1000)] / 30

1

However, discontinuous edges shall be provided with an edge beam with stiffnes ratio, α b, of not less than 0.8, failing which the thickness given by Eq. 1 shall be increased by 10 percent For slabs with drop panels, the minimum thickness of slab is: D ≥ [l (0.6 + f / 1000)] / [30{1+(2x /l )(D -D)/D}]

2

where xd / (ln/2) is smaller of the values determined in the two directions, and xd is not greater than ln/4, and (Dd-D) is not larger than D. For slabs with beams between all supports, the minimum thickness of slab is: D ≥ [ln (0.6 + fy / 1000)] / {30 + 4βαbm }

2a

where αbm is not greater than 2.0. This limit is to ensure that with heavy beams all around the panel, the slab thickness does not become too thin. In the above equations, Dd≡ overall thickness of drop panel, mm; xd ≡ dimension from face of column to edge of drop panel, mm; fy ≡ characteristic yield strength of steel (in MPa); β ≡ (clear long span)/(clear short span); α ≡ average value of α for all beams on edges of slab panel bm

b

α ≡ „beam stiffness parameter‟, defined as the ratio of the flexural stiffness of the beam section to that of b

a width of slab bounded laterally by the centreline of the adjacent panel (if any) on each side of the beam

Referring to Fig. 11.27

α=

3

where I is the second moment of area with respect to the centriodal axis of the gross flanged section of b

3

the beam [shaded area in Fig. 11.27( (a),(b), (c) and (d)) and Is = I2D /12. The value of Ib may be taken as 3

Ib = bD /12 (2.5 (1- )

3a

where b is the width and Db is overall depth of the beam. The minimum thickness for flat slabs obtained from Eq. 11.25 are given in Table 11.2 Table 11.2 Minimum thicknesses for two-way slabs without beams between interior column supports (Eq. 11.25)

Fig. 11.27 Definition of beam section Drop Panels The „drop panel‟ is formed by local thickening of the slab in the neighbourhood of supporting column. Drop panels (or simply, drops) are provided mainly for the purpose of reducing shear stresses around the column supports. They also help in reducing the steel requirement for „negative‟ moments at the column supports. [Also refer Section 11.7 for calculation of reinforcement at drop panels]. The Code (Cl. 31.2.2) recommends that drops should be rectangular in plan, and have a length in each direction not less than one-third† of the panel length in that direction. For the exterior panels, the length, measured perpendicular to the discontinuous edge from the column centreline should be taken as onehalf of the of the corresponding width of drop for the interior panel [Fig. 11.28(a)]. The Code does not specify a minimum thickness required fpr the drop panel except that in computation of negative moment reinforcement, the effective depth of the drop panel shall not exceed tye thickness of the slab plus one quarter the distance between the edge of the drop and edge of the capital (cl 31.7.2). It is, however, recommended [Ref. 11.18, 11.19] that the projection below the slab should not be less than one-fourth the slab thickness, and preferably not less than 100mm [Fig. 11.28(b)].

Column Capital

The „column capital‟ (or column head), provided at the top of a column, is intended primarily to increase the capacity of the slab to resist punching shear [see Section 11.8.2]. The flaring of the column at top is generally done such that the plan geometry at the column head is similar to that of the column. The Code (Cl. 31 2.3) restrict the structurally useful portion of the column capital to that portion which lies within the largest (inverted) pyramid or right circular cone which has a vertex angle of 90 degrees, and can be included entirely within the outlines of the column and the column head [Fig. 11.28(b)]. This is based on the assumption of a 45 degree failure plane, outside of which enlargements of the support are considered ineffective in transferring shear to the column

Fig. 11.28 Drop panel and column capital

TRANSFER OF SHEAR AND MOMENTS TO COLUMNS IN BEAMLESS TWO-WAY SLABS Shear forces and bending moments have to be transferred between the floor system and the supporting columns. In slabs without beams along column lines, this needs special considerations. The design moments in the slabs are computed by frame analysis in the case of the Equivalent Frame Method, and by empirical equations in the case of the Direct Design Method. At any column support, the total

unbalanced moment must be resisted by the columns above and below in proportion to their relative stiffnesses [Fig. 11.29(a)]. In slabs without beams along the column line, the transfer of the unbalanced moment from the slab to the column takes place partly through direct flexural stresses, and partly through development of non-uniform shear stresses around the MMcolumn head. A part (Mub) of the unbalanced moment can be considered to the transferred by flexure and the balance (Muv) through eccentricity of shear forces, as shown in Fig 11.29 (b) and (c). The Code recommendation (Cl. 31.3.3) for the apportioning of M ub and Muv is based on a study described in Ref. 11.23 Mub = γMu Muv = (1-γ)Mu where γ = √

Here c1 and c2 are the dimensions of the equivalent rectangular column, capital or bracket, measured in the direction moments are being determined and in the transverse direction, respectively, and d is the effective depth of the slab at the critical section for shear [refer Section 11.8.2]. For square and round columns, c1 = c2, and γ= = 0.6.

(b) moment transferred by flexure

(c) moment transferred through shear Fig. 11.29 Transfer of unbalanced moment from slab to column The width of the slab considered effective in resisting the moment Mub is taken as the width between lines a distance 1.5 times slab/drop thickness on either side of the column or column capital [Fig. 11.29(b)], and hence this strip should have adequate reinforcement to resist this moment. The detailing of reinforcement for moment transfer, particularly at the exterior column where the unbalanced moment is usually the largest, is critical for the safety as well as the performance of flat slabs without edge beams. The critical section considered for moment transfer by eccentricity of shear is at a distance d/2 from the periphery of the column or column capital [Fig. 11.29(c)]. The shear stresses introduced because of the moment transfer, (assumed to vary linearly about the centroid of the critical section), should be added to the shear stresses due to the vertical support reaction [refer Section 11.8.2].

Fig 2 Load transfer in wall-supported and column-supported slabs

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(a) Two way beam supported slab

(c) Two way beam supFlatported slab

(b) Flat Plate

(d) Waffle Slab

Types of Two-Way Slab Systems

Figure 2 Plans of two-way slabs The absence of beams in flat plates and flat slabs lead to the following advantages. 1) Formwork is simpler 2) Reduced obstruction to service conduits 3) More flexibility in interior layout and future refurbishment. Two-way slabs can be post-tensioned. The main advantage of prestressing a slab is the increased spanto-depth ratio. As per ACI 318-02 (Building Code Requirements for Structural Concrete, American Concrete Institute), the limits of span-to-depth ratios are as follows. For floors 42 For roofs 48.

The values can be increased to 48 and 52, respectively, if the deflection, camber and vibration are not objectionable. The following photographs show post-tensioned flat plate and flat slab.

Analysis The analysis of two-way slabs is given in Section 31, IS:456 - 2000, under “Flat Slabs”.The analysis is applicable to flat plates, flat slabs and two-way slabs with deflecting beams. For two-way slabs with beams, if the beams are sufficiently stiff, then the method (based on moment coefficients) given in Annex D, IS:456 – 2000, is applicable. The direct design method of analysing a two-way slab is not recommended for prestressed slabs. The equivalent frame method is recommended by ACI 318-02. It is given in Subsection 31.5, IS:456 - 2000. This method is briefly covered in this section for flat plates and flat slabs. Prestressed Concrete Structures Dr. Amlan K Sengupta and Prof. Devdas

(a) Flat plate

(b) Flat slab Figure 3 Post-tensioned two-way slabs (Courtesy: VSL India Pvt. Ltd.)

Flat Slabs A reinforced concrete flat slab, also called as beamless slab, is a slab supported directly by columns without beams. A part of the slab bounded on each of the four sides by centre line of column is called panel. The flat slab is often thickened closed to supporting columns to provide adequate strength in shear and to reduce the amount of negative reinforcement in the support regions. The thickened portion i.e. the projection below the slab is called drop or drop panel. In some cases, the section of column at top, as it meets the floor slab or a drop panel, is enlarged so as to increase primarily the perimeter of the critical section, for shear and hence, increasing the capacity of the slab for resisting two-way shear and to reduce negative bending moment at the support. Such enlarged or flared portion of and a capital. Slabs of constant thickness which do not have drop panels or column capitals are referred to as flat plates. The strength of the flat plate structure is often limited due to punching shear action around columns, and consequently they are used for light loads and relatively small spans In general, in column-supported slabs, with or without beams along the column lines, 100 percent of the slab load has to be transmitted by the floor system in both directions (transverse and longitudinal) towards the columns. In such cases, the entire floor system and the columns act integrally in a two-way frame action. Proportioning of Slab Thickness, Drop Panel and Column Head Slab Thickness The thickness of the slab is generally governed by deflection control criteria. [Shear is also an important design criterion — especially in flat plates (slabs without beams and drop panels) and at exterior column supports]. The calculation of deflections of two-way slab systems is quite complex, and recourse is often made to empirical rules which limit maximum span/depth ratios as indirect measures of deflection control. For this purpose, the Code (Cl. 31.2.1) recommends the same l/d ratios prescribed (in Cl. 23.2, also refer Section 5.3.2) for flexural members in general, with the following important differences:

the longer span should be considered‡ (unlike the case of slabs supported on walls or stiff beams, where the shorter span is considered); for the purpose of calculating the modification factor kt [Table 5.2] for tension reinforcement, an average percentage of steel across the whole width of panel should be considered [Ref. 11.11]; When drop panels conforming to Cl. 31.2.2 are not provided around the column supports, in flat slabs the calculated l/d ratios should be further reduced by a factor of 0.9; the minimum thickness of the flat slab should be 125 mm

Slab Thickness Recommended by other Codes Other empirical equations fo um span/depth ratios have been established,based on the results of extensive tests on floor slabs, and have been supported by past experience with such construction under normal values of uniform loading[Ref. 11.18, 11.19]. Thus Ref. 11.8 recommends equations 11.25 to 11.26a for the minimum overall thickness of slabs necessary for the control of deflections. If these minimum thickness requirements are satisfied, deflections need not be computed. These equations are also applicable for two-way slabs supported on stiff beams. However, these thicknesses may not be the most economical in all cases, and may even be inadequate for slabs with large live to dead load ratios. In the calculation of span/depth ratio, the clear span ln in the longer direction and the overall depth (thickness) D are to be considered. For flat plates and slabs with column capitals, the minimum overall thickness of slab is: D ≥ [ln (0.6 + fy / 1000)] / 30

1

However, discontinuous edges shall be provided with an edge beam with stiffnes ratio, α b, of not less than 0.8, failing which the thickness given by Eq. 1 shall be increased by 10 percent For slabs with drop panels, the minimum thickness of slab is: D ≥ [l (0.6 + f / 1000)] / [30{1+(2x /l )(D -D)/D}]

2

where xd / (ln/2) is smaller of the values determined in the two directions, and xd is not greater than ln/4, and (Dd-D) is not larger than D. For slabs with beams between all supports, the minimum thickness of slab is: D ≥ [ln (0.6 + fy / 1000)] / {30 + 4βαbm }

2a

where αbm is not greater than 2.0. This limit is to ensure that with heavy beams all around the panel, the slab thickness does not become too thin. In the above equations, Dd≡ overall thickness of drop panel, mm; xd ≡ dimension from face of column to edge of drop panel, mm; fy ≡ characteristic yield strength of steel (in MPa); β ≡ (clear long span)/(clear short span); α ≡ average value of α for all beams on edges of slab panel bm

b

α ≡ „beam stiffness parameter‟, defined as the ratio of the flexural stiffness of the beam section to that of b

a width of slab bounded laterally by the centreline of the adjacent panel (if any) on each side of the beam

Referring to Fig. 11.27

α=

3

where I is the second moment of area with respect to the centriodal axis of the gross flanged section of b

3

the beam [shaded area in Fig. 11.27( (a),(b), (c) and (d)) and Is = I2D /12. The value of Ib may be taken as 3

Ib = bD /12 (2.5 (1- )

3a

where b is the width and Db is overall depth of the beam. The minimum thickness for flat slabs obtained from Eq. 11.25 are given in Table 11.2 Table 11.2 Minimum thicknesses for two-way slabs without beams between interior column supports (Eq. 11.25)

Fig. 11.27 Definition of beam section Drop Panels The „drop panel‟ is formed by local thickening of the slab in the neighbourhood of supporting column. Drop panels (or simply, drops) are provided mainly for the purpose of reducing shear stresses around the column supports. They also help in reducing the steel requirement for „negative‟ moments at the column supports. [Also refer Section 11.7 for calculation of reinforcement at drop panels]. The Code (Cl. 31.2.2) recommends that drops should be rectangular in plan, and have a length in each direction not less than one-third† of the panel length in that direction. For the exterior panels, the length, measured perpendicular to the discontinuous edge from the column centreline should be taken as onehalf of the of the corresponding width of drop for the interior panel [Fig. 11.28(a)]. The Code does not specify a minimum thickness required fpr the drop panel except that in computation of negative moment reinforcement, the effective depth of the drop panel shall not exceed tye thickness of the slab plus one quarter the distance between the edge of the drop and edge of the capital (cl 31.7.2). It is, however, recommended [Ref. 11.18, 11.19] that the projection below the slab should not be less than one-fourth the slab thickness, and preferably not less than 100mm [Fig. 11.28(b)].

Column Capital

The „column capital‟ (or column head), provided at the top of a column, is intended primarily to increase the capacity of the slab to resist punching shear [see Section 11.8.2]. The flaring of the column at top is generally done such that the plan geometry at the column head is similar to that of the column. The Code (Cl. 31 2.3) restrict the structurally useful portion of the column capital to that portion which lies within the largest (inverted) pyramid or right circular cone which has a vertex angle of 90 degrees, and can be included entirely within the outlines of the column and the column head [Fig. 11.28(b)]. This is based on the assumption of a 45 degree failure plane, outside of which enlargements of the support are considered ineffective in transferring shear to the column

Fig. 11.28 Drop panel and column capital

TRANSFER OF SHEAR AND MOMENTS TO COLUMNS IN BEAMLESS TWO-WAY SLABS Shear forces and bending moments have to be transferred between the floor system and the supporting columns. In slabs without beams along column lines, this needs special considerations. The design moments in the slabs are computed by frame analysis in the case of the Equivalent Frame Method, and by empirical equations in the case of the Direct Design Method. At any column support, the total

unbalanced moment must be resisted by the columns above and below in proportion to their relative stiffnesses [Fig. 11.29(a)]. In slabs without beams along the column line, the transfer of the unbalanced moment from the slab to the column takes place partly through direct flexural stresses, and partly through development of non-uniform shear stresses around the MMcolumn head. A part (Mub) of the unbalanced moment can be considered to the transferred by flexure and the balance (Muv) through eccentricity of shear forces, as shown in Fig 11.29 (b) and (c). The Code recommendation (Cl. 31.3.3) for the apportioning of M ub and Muv is based on a study described in Ref. 11.23 Mub = γMu Muv = (1-γ)Mu where γ = √

Here c1 and c2 are the dimensions of the equivalent rectangular column, capital or bracket, measured in the direction moments are being determined and in the transverse direction, respectively, and d is the effective depth of the slab at the critical section for shear [refer Section 11.8.2]. For square and round columns, c1 = c2, and γ= = 0.6.

(b) moment transferred by flexure

(c) moment transferred through shear Fig. 11.29 Transfer of unbalanced moment from slab to column The width of the slab considered effective in resisting the moment Mub is taken as the width between lines a distance 1.5 times slab/drop thickness on either side of the column or column capital [Fig. 11.29(b)], and hence this strip should have adequate reinforcement to resist this moment. The detailing of reinforcement for moment transfer, particularly at the exterior column where the unbalanced moment is usually the largest, is critical for the safety as well as the performance of flat slabs without edge beams. The critical section considered for moment transfer by eccentricity of shear is at a distance d/2 from the periphery of the column or column capital [Fig. 11.29(c)]. The shear stresses introduced because of the moment transfer, (assumed to vary linearly about the centroid of the critical section), should be added to the shear stresses due to the vertical support reaction [refer Section 11.8.2].

Fig 2 Load transfer in wall-supported and column-supported slabs

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