31flat slabs

31.1.1

For the purpose of this clause, the following definitions shall apply:

• a) Column strip — Column strip means a design strip having a width of 0.25 $l_{2}$, but not greater than $0.25 l_{1}$ on each side of the column centreline, where $l_{1}$ is the span in the direction moments are being determined, measured centre to centre of supports and $l_{2}$ is the span transverse

to $l_{1}$, measured centre to centre of supports.

• b) Middle strip-Middle strip means a design strip bounded on each of its opposite sides by the column strip.
• c) Panel — Panel means that part of a slab bounded on each of its four sides by the centre-line of a column or centre-lines of adjacent spans.

31.2.1 Thickness of Flat Slab

The thickness of the flat slab shall be generally controlled by considerations of span to effective depth ratios given in 23.2.

For slabs with drops conforming to 31.2.2, span to effective depth ratios given in 23.2 shall be applied directly; otherwise the span to effective depth ratios obtained in accordance with provisions in 23.2 shall be multiplied by 0.9. For this purpose, the longer span shall be considered. The minimum thickness of slab shall be 125 mm.

31.2.2 Drop

The drops when provided shall be rectangular in plan, and have a length in each direction not less than one-third of the panel length in that direction. For exterior panels, the width of drops at right angles to the non-continuous edge and measured from the centre-line of the columns shall be equal to one-half the width of drop for interior panels.

31.2.3 Column Heads

Where column heads are provided, that portion of a column head which lies within the largest right circular cone or pyramid that has a vertex angle of 90°and can be included entirely within the outlines of the column and the column head, shall be considered for design purposes (see Fig.12).

31.3.1 Methods of Analysis and Design

It shall be permissible to design the slab system by one of the following methods:

• a) The direct design method as specified in 31.4, and
• b) The equivalent frame method as specified in 31.5.

In each case the applicable limitations given in 31.4 and 31.5 shall be met.

31.3.2 Bending Moments in Panels with Marginal Beams or Walls

Where the slab is supported by a marginal beam with a depth greater than 1.5 times the thickness of the slab, or by a wall, then:

Figure description:

Engineering Diagram: Critical Sections for Shear in Concrete Slabs

Key Variables:

• D: Diameter of column or column head.
• d: Effective depth of slab or drop.
• De: Diameter of the critical section for design.

Configurations & Critical Section Locations:

• 12 A: Slab Without Drop & Column Without Column Head

  • The critical section for shear is located at a distance of d/2 from the face of the column.
  • A 90° angle from the column center helps define the effective area.

• 12 B: Slab With Drop & Column With Column Head

  • Primary critical section: Located at d/2 from the edge of the column head.
  • Secondary critical section: Located at d/2 from the edge of the drop panel.

• 12 C: Slab Without Drop & Column With Column Head

  • The critical section is located at d/2 from the edge of the column head.
  • Constraint: Concrete outside the 90° cone projected from the column center is neglected in calculations.

NOTE — D is the diameter of column or column head to be considered for design and d is effective depth of slab or drop as appropriate.

FIG. 12 CRITICAL SECTIONS FOR SHEAR IN FLAT SLABS

• a) the total load to be carried by the beam or wall shall comprise those loads directly on the wall or beam plus a uniformly distributed load equal to one-quarter of the total load on the slab, and

b) the bending moments on the half-column strip

• The bending moments on the hair column strip adjacent to the beam or wall shall be one-quarter of the bending moments for the first interior column strip.

31.3.3 Transfer of Bending Moments to Columns

When unbalanced gravity load, wind, earthquake, or

When unbalanced gravity load, wind, earthquake, or other lateral loads cause transfer of bending moment between slab and column, the flexural stresses shall be investigated using a fraction, α of the moment given by:

Figure description:

1 απ α=143% 1+

where

• $\begin{aligned} a_{1} & =\text { overall dimension of the critical section } \\ & \text { for shear in the direction in which } \\ & \text { moment acts, and }\end{aligned}$

moment acts, and $\begin{aligned} a_{2} & =\text { overall dimension of the critical section } \\ & \text { for bending, measured to the direction in }\end{aligned}$

• overall dimension of the critical section for shear transverse to the direction in which moment acts.

A slab width between lines that are one and one-half slab or drop panel thickness; 1.5 D, on each side of the column or capital may be considered effective, D being the size of the column.

Concentration of reinforcement over column head by closer spacing or additional reinforcement may be used to resist the moment on this section.

31.4.1 Limitations

Slab system designed by the direct design method shall fulfil the following conditions:

a) There shall be minimum of three continuous

• spans in each direction,
• b) The panels shall be rectangular, and the ratio of the longer span to the shorter span within a panel shall not be greater than 2.0,

c) It shall be permissible to offset columns to a

• It shall be permissible to offset columns to a maximum of 10 percent of the span in the direction of the offset notwithstanding the provision in (b).

d) The successive span lengths in each direction

• The successive span lengths in each direction shall not differ by more than one-third of the longer span. The end spans may be shorter but not longer than the interior spans, and

• e) The design live load shall not exceed three times the design dead load.

31.4.2 Total Design Moment for a Span

31.4.2.1

In the direct design method, the total design moment for a span shall be determined for a strip bounded laterally by the centre-line of the panel on each side of the centre-line of the supports.

31.4.2.2

The absolute sum of the positive and average negative bending moments in each direction shall be taken as:

$M_{\circ}=\frac{w}{8}$ where

$M_{\mathrm{t}}=$ total moment;

• $W=$ design load on an area $l_{2} l_{\mathrm{a}}$ ;
• = clear span extending from face to face of columns, capitals, brackets or walls, but not less than 0.65 $l_{1}$;
• $l_{1}=$ length of span in the direction of $M_{o}$; and
• length of span transverse to $l_{1}$.

31.4.2.3

Circular supports shall be treated as square supports having the same area.

31.4.2.4

When the transverse span of the panels on either side of the centre-line of supports varies, $l_{2}$ shall be taken as the average of the transverse spans.

31.4.2.5

When the span adjacent and parallel to an edge is being considered, the distance from the edge to the centre-line of the panel shall be substituted for $l_{2}$ in 31.4.2.2.

31.4.3 Negative and Positive Design Moments

31.4.3.1

The negative design moment shall be located at the face of rectangular supports, circular supports being treated as square supports having the same area.

31.4.3.2

In an interior span, the total design moment $M$ shall be distributed in the following proportions:

31.4.3.3

In an end span, the total design moment M,

shall be distributed in the following proportions:

Interior negative design moment:

$0.75-\frac{0.10}{1+\frac{1}{\alpha_{2}}}$

Positive design moment:

$0.63-\frac{0.28}{1+\frac{1}{\alpha_{\mathrm{c}}}}$

Exterior negative design moment: 0.65 $1+\frac{1}{\alpha}$

$\alpha_{\mathrm{c}}$ is the ratio of flexural stiffness of the exterior columns to the flexural stiffness of the slab at a joint taken in the direction moments are being determined and is given by

$\alpha_{\mathrm{c}}=\frac{\Sigma K_{\mathrm{c}}}{K_{\mathrm{L}}}$

where

$K_{\mathrm{c}}=$

• sum of the flexural stiffness of the columns meeting at the joint; and
• $K_{\mathrm{s}}=$ flexural stiffness of the slab, expressed as moment per unit rotation.

31.4.3.4

It shall be permissible to modify these design moments by up to 10 percent, so long as the total design moment, $M_{\mathrm{u}}$ for the panel in the direction considered is not less than that required by 31.4.2.2.

31.4.3.5

The negative moment section shall be designed to resist the larger of the two interior negative design moments determined for the spans framing into a common support unless an analysis is made to distribute the unbalanced moment in accordance with the stiffness of the adjoining parts.

31.4.4 Distribution of Bending Moments Across the Panel Width

Bending moments at critical cross-section shall be

distributed to the column strips and middle strips as specified in 31.5.5 as applicable.

31.4.5 Moments in Columns

31.4.5.1

Columns built integrally with the slab system shall be designed to resist moments arising from loads on the slab system.

31.4.5.2

At an interior support, the supporting members above and below the slab shall be designed to resist the moment $M$ given by the following equation, in direct proportion to their stiffnesses unless a general analysis is made:

$M=0.08 \frac{\left(w_{d}+0.5 w_{1}\right) l_{2} l_{\circ}^{2}-w}{1+\frac{1}{-}}$

where

• $w_{d}, w_{l}=$ design dead and live loads respectively, per unit area;

• $\begin{aligned} l_{2} & =\text { length of span transverse to the } \\ & \text { direction of } M ;\end{aligned}$

• = length of the clear span in the direction of M, measured face to face of supports;

ΣK

$\alpha_{\mathrm{c}}=\frac{K_{\mathrm{c}}}{\Sigma K_{\mathrm{s}}}$ where $K_{\mathrm{c}}$ and $K_{\mathrm{s}}$ are as defined in 31.4.3.3; and

$w_{a}^{\prime}, l_{2}^{\prime}$ and $l_{a}^{\prime}$, refer to the shorter span.

31.4.6 Effects of Pattern Loading

In the direct design method, when the ratio of live load to dead load exceeds 0.5 :

• a) the sum of the flexural stiffnesses of the columns above and below the slab, ΣKc, shall be such that αc is not less than the appropriate minimum value αc min specified in Table 17, or
• b) if the sum of the flexural stiffnesses of the columns, ΣKc, does not satisfy (a), the positive design moments for the panel shall be multiplied by the coefficient β, given by the following equation:

 _Figure description:_ The image shows a mathematical formula for calculating the value **$\beta_s$**: \right.\beta_s = 1 + \left( \frac{2 - \frac{w_d}{w_l}}{4 + \frac{w_d}{w_l}} \right \left( 1 - \frac{\alpha_c}{\alpha_{c \text{min}}} \right$$

Key Entities:

• $\beta_s$: The result of the calculation.
• $w_d / w_l$: Ratio of dead load to live load.
• $\alpha_c$: Ratio of flexural stiffness of columns above and below a slab to the flexural stiffness of the slabs at a joint.
• $\alpha_{c \text{ min}}$: A minimum reference value for the stiffness ratio.

$\alpha_{\mathrm{c}}$ is the ratio of flexural stiffness of the columns above and below the slab to the flexural stiffness of the slabs at a joint taken in the direction moments are being determined and is given by:

• $\alpha_{\mathrm{c}}=\frac{\Sigma K_{\mathrm{c}}}{\Sigma K_{\mathrm{s}}}$

where $K_{\mathrm{c}}$ and $K_{\mathrm{s}}$ are flexural stiffnesses of column and slab respectively.

31.5.1 Assumptions

The bending moments and shear forces may be determined by an analysis of the structure as a continuous frame and the following assumptions may be made:

• a) The structure shall be considered to be made up of equivalent frames on column lines taken longitudinally and transversely through the building. Each frame consists of a row of equivalent columns or supports, bounded laterally by the centre-line of the panel on each side of the centre-line of the columns or supports. Frames adjacent and parallel to an edge shall be bounded by the edge and the centre-line of the adjacent panel.
• b) Each such frame may be analyzed in its entirety, or, for vertical loading, each floor thereof and the roof may be analyzed separately with its columns being assumed fixed at their remote ends. Where slabs are thus analyzed separately, it may be assumed in determining the bending moment at a given support that the slab is fixed at any support two panels distant therefrom provided the slab continues beyond the point.
• c) For the purpose of determining relative stiffness of members, the moment of inertia of any slab or column may be assumed to be that of the gross cross-section of the concrete alone.
• d) Variations of moment of inertia along the axis of the slab on account of provision of drops shall be taken into account. In the case of recessed or coffered slab which is made solid in the region of the columns, the stiffening effect may be ignored provided the solid part of the slab does not extend more than 0.15 $l_{\mathrm{cr}}$, into the span measured from the centre-line of the columns. The stiffening effect of flared column heads may be ignored.

31.5.2 Loading Pattern

31.5.2.1

When the loading pattern is known, the structure shall be analyzed for the load concerned.

31.5.2.2

When the live load is variable but does not exceed three-quarters of the dead load, or the nature of the live load is such that all panels will be loaded simultaneously, the maximum moments may be assumed to occur at all sections when full design live load is on the entire slab system.

31.5.2.3

For other conditions of live load/dead load ratio and when all panels are not loaded simultaneously:

• a) maximum positive moment near midspan of a panel may be assumed to occur when three-quarters of the full design live load is on the panel and on alternate panels; and
• b) maximum negative moment in the slab at a support may be assumed to occur when three-quarters of the full design live load is on the adjacent panels only.

31.5.2.4

In no case shall design moments be taken to be less than those occurring with full design live load on all panels.

31.5.3 Negative Design Moment

31.5.3.1

At interior supports, the critical section for negative moment, in both the column strip and middle strip, shall be taken at the face of rectilinear supports, but in no case at a distance greater than 0.175 $l_{1}$ from the centre of the column where $l_{1}$ is the length of the span in the direction moments are being determined, measured centre-to-centre of supports.

31.5.3.2

At exterior supports provided with brackets or capitals, the critical section for negative moment in the direction perpendicular to the edge shall be taken at a distance from the face of the supporting element not greater than one-half the projection of the bracket or capital beyond the face of the supporting element.

31.5.3.3

Circular or regular polygon shaped supports shall be treated as square supports having the same area.

31.5.4 Modification of Maximum Moment

Moments determined by means of the equivalent frame method, for slabs which fulfil the limitations of 31.4 may be reduced in such proportion that the numerical sum of the positive and average negative moments is not less than the value of total design moment $M_{o}$ specified in 31.4.2.2.

31.5.5 Distribution of Bending Moment Across the Panel Width

31.5.5.1 Column strip: Negative moment at an interior support

At an interior support, the column strip shall be designed to resist 75 percent of the total negative moment in the panel at that support.

31.5.5.2 Column strip: Negative moment at an exterior support

• a) At an exterior support, the column strip shall be designed to resist the total negative moment in the panel at that support.
• b) Where the exterior support consists of a column or a wall extending for a distance equal to or

greater than three-quarters of the value of $l_{2}$, the length of span transverse to the direction moments are being determined, the exterior negative moment shall be considered to be uniformly distributed across the length $l_{2}$.

31.5.5.3 Column strip: Positive moment for each span

For each span, the column strip shall be designed to resist 60 percent of the total positive moment in the panel.

31.5.5.4 Moments in the middle strip

The middle strip shall be designed on the following bases:

• a) That portion of the design moment not resisted by the column strip shall be assigned to the adjacent middle strips.
• b) Each middle strip shall be proportioned to resist the sum of the moments assigned to its two half middle strips.
• c) The middle strip adjacent and parallel to an edge supported by a wall shall be proportioned to resist twice the moment assigned to half the middle strip corresponding to the first row of interior columns.

31.6.1

The critical section for shear shall be at a distance d/2 from the periphery of the column/capital/ drop panel, perpendicular to the plane of the slab where d is the effective depth of the section (see Fig. 12). The shape in plan is geometrically similar to the support immediately below the slab (see Fig. 13A and 13B).

NOTE — For column sections with re-entrant angles, the critical section shall be taken as indicated in Fig. 13C and 13D.

31.6.1.1

In the case of columns near the free edge of a slab, the critical section shall be taken as shown in Fig. 14.

31.6.1.2

When openings in flat slabs are located at a distance less than ten times the thickness of the slab from a concentrated reaction or when the openings are located within the column strips, the critical sections specified in 31.6.1 shall be modified so that the part of the periphery of the critical section which is enclosed by radial projections of the openings to the centroid of the reaction area shall be considered ineffective (see Fig. 15), and openings shall not encroach upon column head.

31.6.2 Calculation of Shear Stress

The shear stress τv shall be the sum of the values calculated according to 31.6.2.1 and 31.6.2.2.

31.6.2.1

The nominal shear stress in flat slabs shall be taken as $V / b_{\mathrm{o}} d$ where $V$ is the shear force due to design load, $b_{\mathrm{o}}$ is the periphery of the critical section and $d$ is the effective depth.

Figure description:

Key Entities and Information:

• Subject: Structural diagrams illustrating critical sections for shear in flat slabs (Figs. 13 and 14.
• Support Section: Shaded areas representing columns or column heads of various shapes (rectangular, circular, cross-shaped, and L-shaped.
• Critical Section: Indicated by dashed lines, this represents the perimeter where shear strength is evaluated.
• Offset Dimension (d/2: The critical section is located at a distance of half the effective depth ($d/2$ from the face of the support.
• Effect of Boundaries:
  • Free Edge (14A: The critical section is truncated by the slab boundary.
  • Free Corner (14B: The critical section is truncated by two perpendicular slab boundaries at a corner column.
• Key Variable: $d$ = effective depth of the flat slab or drop.

FIG. 14 EFFECT OF FREE EDGES ON CRITICAL SECTION FOR SHEAR

31.6.2.2

When unbalanced gravity load, wind, earthquake or other forces cause transfer of bending moment between slab and column, a fraction $(1-\alpha)$ of the moment shall be considered transferred by eccentricity of the shear about the centroid of the critical section. Shear stresses shall be taken as varying linearly about the centroid of the critical section. The

value of $\alpha$ shall be obtained from the equation given in 31.3.3.

31.6.3 Permissible Shear Stress

31.6.3.1

When shear reinforcement is not provided, the calculated shear stress at the critical section shall not exceed $k_{s} \tau_{\mathrm{c}}$.

IS 4~6: 2000

Figure description:

Key Entities and Information:

• Column: Central square element representing a structural support.
• Critical Section: Indicated by a dashed line perimeter located at a distance of d/2 from the column faces.
• Opening: A void area located near the column.
• Periphery Deduction: The diagram illustrates that the portion of the critical section enclosed by radial lines from the column's center to the edges of the opening must be subtracted from the total design periphery.
• Dimension: The distance from the column face to the critical section is defined as half the effective depth (d/2.

Figure description:

• Column: A central shaded rectangular structural element.
• Critical Section: A dashed perimeter surrounding the column, used for shear calculations.
• Opening: A square area with an 'X' located near the column.
• Dimension d/2: The offset distance from each column face to the critical section boundary.
• Influence Lines: Lines extending from the column center through the opening edges, indicating a reduction in the effective critical section perimeter.
• Circular Feature: A circular element with an 'X' at the bottom, also within the field of influence from the column center.

Figure description:

Fig. 15: Effect of Openings on Critical Section for Shear

Key Entities:

• Column: Represented as a stippled square.
• Critical Section: Indicated by a dashed perimeter around the column.
• Opening: A square or rectangular area (marked with an 'X' that reduces the effective perimeter of the critical section.

Key Details:

• 15 A (Small Opening: Part of the critical section perimeter is considered ineffective if an opening is nearby. The excluded portion is defined by radial lines extending from the column center to the edges of the opening.
• 15 B (Large Opening: A large opening adjacent to the column is treated as a "free edge," causing the critical section perimeter to be truncated at the opening's boundary.

FIG. 15 EFFECT OF OPENINGS ON CRITICAL SECTION FOR SHEAR

where

• $k_{\mathrm{s}}=\left(0.5+\beta_{\mathrm{c}}\right)$ but not greater than $1, \beta_{\mathrm{c}}$ being the ratio of short side to long side of the column/ capital; and
• $=0.25 \sqrt{f_{\mathrm{ck}}}$ in limit state method of design,

and $0.16 \sqrt{f_{\mathrm{ck}}}$ in working stress method of design.

31.6.3.2

When the shear stress at the critical section exceeds the value given in 31.6.3.1, but less than 1.5 $\tau_{\mathrm{c}}$ shear reinforcement shall be provided. If the shear stress exceeds $1.5 \tau_{\mathrm{c}}$, the flat slab shall be redesigned. Shear stresses shall be investigated at successive sections more distant from the support and shear reinforcement shall be provided up to a section where the shear stress does not exceed $0.5 \tau_{\mathrm{c}}$. While designing the shear reinforcement, the shear stress carried by the concrete shall be assumed to be $0.5 \tau_{\mathrm{c}}$ and reinforcement shall carry the remaining shear.

31.7.1 Spacing

The spacing of bars in a flat slab, shall not exceed

2 times the slab thickness, except where a slab is of

2 times the slab thickness, except where a slab is of cellular or ribbed construction.

31.7.2 Area of Reinforcement

When drop panels are used, the thickness of drop panel for determination of area of reinforcement shall be the lesser of the following:

• a) Thickness of drop, and

b) Thickness of slab plus one quarter the distance

• between edge of drop and edge of capital.

31.7.3 Minimum Length of Reinforcement

a) Reinforcement in flat slabs shall have the

• minimum lengths specified in Fig.16. Larger lengths of reinforcement shall be provided when required by analysis.

b) Where adjacent spans are unequal, the extension

• where high overhangs are present, the extension of negative reinforcement beyond each face of the common column shall be based on the longer span.

c) The length of reinforcement for slabs in frames

• The length of reinforcement for slabs in frames not braced against sideways and for slabs resisting lateral loads shall be determined by analysis but shall not be less than those prescribed in Fig. 16.

Figure description:

• ℓn: Clear span.
• Supports: Exterior support (no slab continuity and Interior support (continuity provided.
• D: Distance from face of support.

• Bent bars at exterior supports may be used if a general analysis is made.

NOTE — D is the diameter of the column and the dimension of the rectangular column in the direction under consideration.

FIG. 16 MINIMUM BEND JOINT LOCATIONS AND EXTENSIONS FOR REINFORCEMENT

IN FLAT SLABS

31.7.4 Anchoring Reinforcement

• a) All slab reinforcement perpendicular to a discontinuous edge shall have an anchorage (straight, bent or otherwise anchored) past the internal face of the spandrel beam, wall or column, of an amount:

1. For positive reinforcement — not less than 150 mm except that with fabric reinforcement having a fully welded transverse wire directly over the support, it shall be permissible to reduce this length to one-half of the width of the support or 50 mm, whichever is greater; and
2. For negative reinforcement — such that the design stress is developed at the internal face, in accordance with Section 3.

• b) Where the slab is not supported by a spandrel beam or wall, or where the slab cantilevers beyond the support, the anchorage shall be obtained within the slab.